research article the dynamic programming method of
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 958920 14 pageshttpdxdoiorg1011552013958920
Research ArticleThe Dynamic Programming Method of Stochastic DifferentialGame for Functional Forward-Backward Stochastic System
Shaolin Ji1 Chuanfeng Sun2 and Qingmeng Wei2
1 Institute for Financial Studies and Institute of Mathematics Shandong University Jinan 250100 China2 Institute of Mathematics Shandong University Jinan 250100 China
Correspondence should be addressed to Shaolin Ji jslsdueducn
Received 23 October 2012 Accepted 18 December 2012
Academic Editor Guangchen Wang
Copyright copy 2013 Shaolin Ji et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper is devoted to a stochastic differential game (SDG) of decoupled functional forward-backward stochastic differentialequation (FBSDE) For our SDG the associated upper and lower value functions of the SDG are defined through the solutionof controlled functional backward stochastic differential equations (BSDEs) Applying the Girsanov transformation methodintroduced by Buckdahn and Li (2008) the upper and the lower value functions are shown to be deterministic We also generalizethe Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations to the path-dependent ones By establishing the dynamic programmingprincipal (DPP) we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper andthe lower path-dependent HJBI equations respectively
1 Introduction
The theory of backward stochastic differential equations(BSDEs) has been studied widely since Pardoux and Peng[1] first introduced the nonlinear BSDEs in 1990 BSDEshave got applications in many fields such as stochasticcontrol (see Peng [2]) stochastic differential games (SDGs)(see Hamadene and Lepeltier [3] Hamadene et al [4])mathematical finance (see El Karoui et al [5]) and partialdifferential equation (PDE) theory (see Peng [6 7]) and soforth
In the aspect of finance the BSDE theory presents a sim-ple formulation of stochastic differential utilities introducedby Duffie and Epstein [8] When the generator 119892 of a BSDEdoes not depend on 119911 the solution 119884 is just the comparisontheorem for BSDEs we know
recursive utility presented in [8] From the view of BSDEby studying some important properties (such as comparisontheorem) of BSDEs El Karoui et al [5] gave the more generalclass of recursive utilities and their properties And later therecursive optimal control problems whose cost functionalsare described by the solution of BSDEare studiedwidely Peng
[7] obtained the Bellmanrsquos dynamic programming principle(DPP) for this kind of problem and proved the value functionto be a viscosity solution of one kind of quasi-linear secondorder PDE that is Hamilton-Jacobi-Bellman (HJB) equationLater for the recursive optimal control problem introducedby a BSDE under Markovian framework by introducing thenotion of backward semigroup of BSDE in Peng [9] theBellmanrsquos DPP is derived and the value function is proved tobe a viscosity solution of a generalized HJB equation
By now the DPP with related HJB equation has becomea powerful approach to solving optimal control and gameproblems (see [10ndash13]) In [10] Buckdahn and Li studieda recursive SDG problem and interpreted the relationshipbetween the controlled system and the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation A point is worthy to men-tion in order to derive the DPP they introduced a Girsanovtransformation method to prove the value functions aredeterministic which is different from the method developedin Peng [9]
There really exist some systems which are modeled onlyby stochastic systems whose evolutions depend on the pasthistory of the states Based on this phenomenon Ji and Yang
2 Mathematical Problems in Engineering
[14] investigated a controlled system governed by a functionalforward-backward stochastic differential equation (FBSDE)and proved the value function to be the viscosity solution ofthe related path-dependent HJB equation
In this paper inspired by [10 14] we will investigatethe SDG problems of the functional FBSDEs Preciselythe dynamics of our SDGs is described by the followingfunctional SDE
119889119883120574119905119906119907
(119904) = 119887 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119904
+ 120590 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]
119883120574119905119906119907
119905= 120574119905
(1)
And the cost functional 119869(120574119905 119906 119907) is defined as119884120574119905119906119907
(119905)whichis the solution of the following functional BSDE
119889119884120574119905119906119907
(119904) = minus 119891 (119883120574119905119906119907
119904 119884
120574119905119906119907
(119904) 119885120574119905119906119907
(119904)
119906 (119904) 119907 (119904)) 119889119904 + 119885120574119905119906119907
(119904) 119889119861 (119904)
119884120574119905119906119907
(119879) = Φ (119883120574119905119906119907
119879) 119904 isin [119905 119879]
(2)
where 120574119905 is a path on [0 119905] The driver 119891 and Φ canbe interpreted as the running cost and the terminal costrespectively Also they depend on the past history of thedynamics Equations (1) and (2) compose a decoupled func-tional FBSDE The concrete conditions on 119887 120590 119891 Φ areshown in the later section
In the context we adopt the strategy against control formThe cost functional 119869(120574119905 119906 119907) is explained as a payoff forplayer I and as a cost for player II The aim of this paper isto show the following lower and upper value functions
119882(120574119905) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906))
119880 (120574119905) = esssup120572isinA119905119879
essinf119907isinV119905119879
119869 (120574119905 120572 (119907) 119907)
(3)
are the viscosity solutions of the following path-dependentHJBI equations respectively
119863119905119882(120574119905) + sup119906isin119880
inf119907isin119881
119867(120574119905119882119863119909119882119863119909119909119882119906 119907) = 0
119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ
119863119905119880(120574119905) + inf119907isin119881
sup119906isin119880
119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907) = 0
119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ
(4)
where
119867(120574119905 119910 119901 119883 119906 119907) =1
2tr (120590120590119879 (120574119905 119906 119907)119883)
+ 119901 sdot 119887 (120574119905 119906 119907)
+ 119891 (120574119905 119910 119901 sdot 120590 (120574119905 119906 119907) 119906 119907)
(5)
where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of
119889 times 119889 symmetric matrices)To solve the above SDG we need the functional Itorsquos
calculus and path-dependent PDEs which are recently intro-duced by Dupire [15] (for a recent account of this theorythe reader may consult [16ndash18] And under the framework offunctional Itorsquos calculus for the non-Markovian BSDEs PengandWang [19] derived a nonlinear Feynman-Kac formula forclassical solutions of path-dependent PDEs For the furtherdevelopment the readers may refer to [20 21])
In this paper we apply the Girsanov transformationmethod in Buckdahn and Li [10] to prove the determinacyof the value functions which is different from the methodintroduced by Peng [7 9] Making use of this methodand the functional Itorsquos calculus (introduced by Dupire [15]and developed by Cont and Fournie [16ndash18]) we completethe study of the zero-sum two-player SDGs in the non-Markovian case and present the lower and upper valuefunctions of our SDG are the viscosity solutions of thecorresponding path-dependent HJBI equations respectively
Different from the HJBI equations developed for stochas-tic delay systems we establish the DPP and derive the HJBIequation in the new framework of functional Itorsquos calculus
This paper is organized as follows Section 2 recalls thefunctional Itorsquos calculus and the well-known results of BSDEswe will use later In Section 3 we formulate our SDGs andget the corresponding DPP Based on the obtained DPP inSection 4 we derive the main result of the paper the lowerand upper value functions are the viscosity solutions of theassociated path-dependent HJBI equations respectively Andwe add the proof for the DPP in the appendix
2 Preliminaries
21 Functional Itorsquos Calculus We present some preliminariesfor functional Itorsquos calculus introduced firstly by Dupire [15]Here we follow the notations in [15]
Let 119879 gt 0 be fixed For each 119905 isin [0 119879] we denote Λ 119905 theset of cadlag functions from [0 119905] to R119889
For 120574 isin Λ 119879 denote 120574(119904) by the value of 120574 at time 119904 isin [0 119879]Thus 120574 = (120574(119904))0le119904le119879 is a cadlag process on [0 119879] and its valueat time 119904 is 120574(119904) 120574119905 = (120574(119904))0le119904le119905 isin Λ 119905 is the path of 120574 upto time 119905 We denote Λ = ⋃
119905isin[0119879]Λ 119905 For each 120574119905 isin Λ and
119909 isin R119889 120574119905(119904) is denoted by the value of 120574119905 at 119904 isin [0 119905] and120574119909
119905= (120574119905(119904)0le119904lt119905 120574119905(119905) + 119909) which is also an element in Λ 119905We now introduce a distance onΛ Let ⟨sdot sdot⟩ and | sdot| denote
the inner product and norm inR119889 For each 0 le 119905 119905 le 119879 and120574119905 120574119905 isin Λ we set
10038171003817100381710038171205741199051003817100381710038171003817 = sup
119904isin[0119905]
1003816100381610038161003816120574119905 (119904)1003816100381610038161003816
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 = sup119904isin[0119905or119905]
1003816100381610038161003816120574119905 (119904 and 119905) minus 120574119905(119904 and 119905)
1003816100381610038161003816
119889infin (120574119905 120574119905) = sup119904isin[0119905or119905]
1003816100381610038161003816120574119905 (119904 and 119905) minus 120574119905(119904 and 119905)
1003816100381610038161003816 +1003816100381610038161003816119905 minus 119905
1003816100381610038161003816
(6)
It is obvious that Λ 119905 is a Banach space with respect to sdot Since Λ is not a linear space 119889infin is not a norm
Mathematical Problems in Engineering 3
Definition 1 A functional 119906 Λ 997891rarr R is Λ-continuous at120574119905 isin Λ if for any 120576 gt 0 there exists 120575 gt 0 such that for each120574119905isin Λ with 119889infin(120574119905 120574119905) lt 120575 one has |119906(120574119905) minus 119906(120574
119905)| lt 120576
119906 is said to be Λ-continuous if it is Λ-continuous at each120574119905 isin Λ
Definition 2 Let 119907 Λ 997891rarr R and 120574119905 isin Λ be given If thereexists 119901 isin R119889 such that
119907 (120574119909
119905) = 119907 (120574119905) + ⟨119901 119909⟩ + 119900 (|119909|) as 119909 997888rarr 0 119909 isin R
119889
(7)
Then we say that 119907 is (vertically) differentiable at 119905 and denotethe gradient of 119863119909119907(120574119905) = 119901 sdot 119907 is said to be verticallydifferentiable in Λ if 119863119909119907(120574119905) exists for each 120574119905 isin Λ TheHessian119863119909119909119907(120574119905) can be defined similarly It is an 119878(119889)-valuedfunction defined on Λ where 119878(119889) is the space of all 119889 times 119889
symmetric matrices
For each 120574119905 isin Λ we denote 120574119905119904(119903) = 120574119905(119903)1[0119905)(119903) +
120574119905(119905)1[119905119904](119903) 119903 isin [0 119904] It is clear that 120574119905119904 isin Λ 119904
Definition 3 For a given 120574119905 isin Λ if one has
119907 (120574119905119904) = 119907 (120574119905) + 119886 (119904 minus 119905) + 119900 (|119904 minus 119905|) as 119904 997888rarr 119905 119904 ge 119905
(8)
then we say that 119907(120574119905) is (horizontally) differentiable in 119905
at 120574119905 and denote 119863119905119907(120574119905) = 119886 119907 is said to be horizontallydifferentiable in Λ if119863119905119907(120574119905) exists for each 120574119905 isin Λ
Definition 4 Define C119895119896(Λ) as the set of function 119907 =
(119907(120574119905))120574119905isinΛ defined on Λ which are 119895 times horizontally and
119896 times vertically differentiable in Λ such that all thesederivatives are Λ-continuous
The following is about the functional Itorsquos formula whichwas firstly obtained by Dupire [15] and then developed byCont and Fournie [18] for a more general formulation
Theorem 5 (functional Itorsquos formula) Let (ΩF (F119905)119905isin[0119879]
119875) be a probability space if 119883 is a continuous semimartingaleand 119907 is in C12
(Λ) then for any 119905 isin [0 119879)
119907 (119883119905) minus 119907 (1198830) = int
119905
0
119863119904119907 (119883119904) 119889119904 + int
119905
0
119863119909119907 (119883119904) 119889119883 (119904)
+1
2int
119905
0
119863119909119909119907 (119883119904) 119889 ⟨119883⟩ (119904) 119875-as
(9)
22 BSDEs In this section we recollect some importantresults which will be used in our SDG problems
Let (ΩF 119875) be the Wiener space where Ω is the set ofcontinuous functions from [0 119879] to R119889 starting from 0 (Ω =
1198620([0 119879]R119889)) 119879 is an arbitrarily fixed real time horizonF
is the completed Borel 120590-algebra overΩ and 119875 is the Wienermeasure Let 119861 be the canonical process 119861(120596 119904) = 120596119904 119904 isin
[0 119879] 120596 isin Ω We denote by F = F119904 0 le 119904 le 119879 the natural
filtration generated by 119861(119905)119905ge0 and augmented by all 119875-nullsets that is F119904 = 120590119861(119903) 119903 le 119904 or N119875 119904 isin [0 119879] F119904
119905=
120590(119861(119903)minus119861(119905) 119905 le 119903 le 119904)orN119875 whereN119875 is the set of all119875-nullsubsets First we present two spaces of processes as follows
S2(0 119879R
119899) = (120595(119905))
0le119905le119879R
119899-valued
F-adapted continuous process
119864 [ sup0le119905le119879
1003816100381610038161003816120595 (119905)10038161003816100381610038162] lt +infin
H2(0 119879R
119899) = (120595(119905))
0le119905le119879
R119899-valued
F-progressively measurable process
100381710038171003817100381712059510038171003817100381710038172= 119864[int
119879
0
1003816100381610038161003816120595 (119905)10038161003816100381610038162119889119905] lt +infin
(10)
Consider 119892 Ω times [0 119879] times R times R119889rarr R for every (119910 119911)
in R times R119889 (119892(119905 119910 119911))119905isin[0119879] is progressively measurable andsatisfies the following conditions
(A1) there exists a constant 119862 ge 0 such that 119875-as for all119905 isin [0 119879] 119910
1 1199102 isin R 1199111 1199112 isin R119889
1003816100381610038161003816119892 (119905 1199101 1199111) minus 119892 (119905 1199102 1199112)1003816100381610038161003816 le 119862 (
10038161003816100381610038161199101 minus 11991021003816100381610038161003816 +
10038161003816100381610038161199111 minus 11991121003816100381610038161003816) (11)
(A2) 119892(sdot 0 0) isin H2(0 119879R)
In the following we suppose the driver 119892 of a BSDEsatisfies (A1) and (A2)
Lemma 6 Under the assumptions (A1) and (A2) for anyrandom variable 120585 isin 119871
2(ΩF119879 119875R) the BSDE
119910 (119905) = 120585 + int
119879
119905
119892 (119904 119910 (119904) 119911 (119904)) 119889119904 minus int
119879
119905
119911 (119904) 119889119861 (119904)
0 le 119905 le 119879
(12)
has a unique adapted solution
(119910(119905) 119911(119905))119905isin[0119879]
isin S2(0 119879R) timesH
2(0 119879R
119889) (13)
The readers may refer to Pardoux and Peng [1] forthe above well-known existence and uniqueness results onBSDEs
Lemma 7 (comparison theorem) Given two coefficients 11989211198922 satisfying (A1) (A2) and two terminal values 1205851 1205852 isin
1198712(ΩF119879 119875R) by (119910
1 119911
1) and (119910
2 119911
2) one denotes the
solution of a BSDE with the data (1205851 1198921) and (1205852 1198922)respectively Then one has the following
(i) If 1205851 ge 1205852 and 1198921 ge 1198922 P-as then 1199101(119905) ge 119910
2(119905)
P-as for all 119905 isin [0 119879]
4 Mathematical Problems in Engineering
(ii) (Strict monotonicity) If in addition to (i) one alsoassumes that 119875(1205851 gt 1205852) gt 0 then 119875(119910
1(119905) gt 119910
2(119905)) gt
0 0 le 119905 le 119879 and in particular 1199101(0) gt 119910
2(0)
With the notations in Lemma 7 we assume that for some119892 Ω times [0 119879] times R times R119889
rarr R satisfying (A1) and (A2) thedrivers 119892119894 have the following form
119892119894 (119904 119910119894(119904) 119911
119894(119904)) = 119892 (119904 119910
119894(119904) 119911
119894(119904)) + 120593119894 (119904)
119889119904119889119875-ae 119894 = 1 2
(14)
where 120593119894 isin H2(0 119879R) Then the following results hold true
for all terminal values 1205851 1205852 isin 1198712(ΩF119879 119875R)
Lemma 8 The difference of the solutions (1199101 119911
1) and (119910
2 119911
2)
of BSDE (12) with the data (1205851 1198921) and (1205852 1198922) respectivelysatisfies the following estimate
100381610038161003816100381610038161199101(119905)minus119910
2(119905)
10038161003816100381610038161003816
2
+1
2119864 [int
119879
119905
119890120573(119904minus119905)
(100381610038161003816100381610038161199101(119904)minus119910
2(119904)
10038161003816100381610038161003816
2
+100381610038161003816100381610038161199111(119904) minus 119911
2(119904)
10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864 [119890120573(119879minus119905)10038161003816100381610038161205851 minus 1205852
10038161003816100381610038162| F119905]
+ 119864 [int
119879
119905
119890120573(119904minus119905)10038161003816100381610038161205931 (119904) minus 1205932 (119904)
10038161003816100381610038162119889119904 | F119905]
P-as forall0 le 119905 le 119879 where 120573 = 16 (1 + 1198622)
(15)
Proof The reader may refer to Proposition 21 in El Karoui etal [5] or Theorem 23 in Peng [9] for the details
3 A DPP for Stochastic Differential Games ofFunctional FBSDEs
In this section we consider the SDGs of functional FBSDEsFirst we introduce the background of SDGs Suppose
that the control state spaces 119880119881 are compact metric spacesU (resp V) is the control set of all 119880 (resp 119881)-valuedF-progressively measurable processes for the first (respsecond) player If 119906 isin U (resp 119907 isin V) we call 119906 (resp 119907)an admissible control
Let us give the following mappings
119887 Λ times 119880 times 119881 997888rarr R119899
120590 Λ times 119880 times 119881 997888rarr R119899times119889
119891 Λ timesR timesR119889times 119880 times 119881 997888rarr R
(16)
For given admissible controls 119906(sdot) isin U 119907(sdot) isin V and 119905 isin
[0 119879] 120574119905 isin Λ we consider the following functional forward-backward stochastic system
119889119883120574119905119906119907
(119904) = 119887 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119904
+ 120590 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]
119889119884120574119905119906119907
(119904) = minus119891 (119883120574119905119906119907
119904 119884
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
119883120574119905119906119907
119905= 120574119905
119884120574119905119906119907
(119879) = Φ (119883120574119905119906119907
119879)
(17)
(H) (i) For all 119905 isin [0 119879] 119906 isin 119880 119907 isin 119881 119909119905 isin Λ 119910 isin
R 119911 isin R119889 119887(119909119905 119906 119907) 120590(119909119905 119906 119907) and 119891(119909119905 119910 119911 119906 119907)
areF119905-measurable(ii) There exists a constant 119862 gt 0 such that for all 119905 isin
[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091
119905 119909
2
119905isin Λ
10038161003816100381610038161003816119887 (119909
1
119905 119906 119907) minus 119887 (119909
2
119905 119906 119907)
10038161003816100381610038161003816+10038161003816100381610038161003816120590 (119909
1
119905 119906 119907) minus 120590 (119909
2
119905 119906 119907)
10038161003816100381610038161003816
le 119862100381710038171003817100381710038171199091
119905minus 119909
2
119905
10038171003817100381710038171003817
(18)1003816100381610038161003816119887 (119909119905 119906 119907)
1003816100381610038161003816 +1003816100381610038161003816120590 (119909119905 119906 119907)
1003816100381610038161003816 le 119862 (1 +1003817100381710038171003817119909119905
1003817100381710038171003817) for any 119909119905 isin Λ
(19)
(iii) There exists a constant 119862 gt 0 such that for all 119905 isin
[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091
119905 119909
2
119905isin Λ 119910
1 119910
2isin
R 1199111 119911
2isin R119889
10038161003816100381610038161003816119891 (119909
1
119905 119910
1 119911
1 119906 119907) minus 119891 (119909
2
119905 119910
2 119911
2 119906 119907)
10038161003816100381610038161003816
le 119862 (100381710038171003817100381710038171199091
119905minus 119909
2
119905
10038171003817100381710038171003817+100381610038161003816100381610038161199101minus 119910
210038161003816100381610038161003816+100381610038161003816100381610038161199111minus 119911
210038161003816100381610038161003816)
10038161003816100381610038161003816Φ (119909
1
119879) minus Φ (119909
2
119879)10038161003816100381610038161003816le 119862
100381710038171003817100381710038171199091
119879minus 119909
2
119879
10038171003817100381710038171003817
1003816100381610038161003816119891 (119909119905 0 0 119906 119907)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171199091199051003817100381710038171003817)
1003816100381610038161003816Φ (119909119879)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171199091198791003817100381710038171003817) for any 119909119905 isin Λ
(20)
Theorem 9 Under the assumption (H) there exists a uniquesolution (119883 119884 119885) isin S2
(0 119879R119899) timesS2
(0 119879R) timesH2(0 119879R119889
)
solving (17)
We recall the subspaces of admissible controls and thedefinitions of admissible strategies as follows which aresimilar to [10]
Definition 10 An admissible control process 119906 = (119906119903)119903isin[119905119904]
(resp 119907 = (119907119903)119903isin[119905119904]) for Player I (resp II) on [119905 119904] is anF119903-progressively measurable 119880 (resp 119881)-valued process Theset of all admissible controls for Player I (resp II) on [119905 119904]
is denoted byU119905119904 (respV119905119904) If 119875119906 equiv 119906 119886119890 119894119899 [119905 119904] = 1one will identify both processes 119906 and 119906 inU119905119904 Similarly oneinterprets 119907 equiv 119907 on [119905 119904] inV119905119904
Mathematical Problems in Engineering 5
Definition 11 A nonanticipative strategy for Player I on[119905 119904] (119905 lt 119904 le 119879) is a mapping 120572 V119905119904 rarr U119905119904 suchthat for any F-stopping time 119878 Ω rarr [119905 119904] and any1199071 1199072 isin V119905119904 with 1199071 equiv 1199072 on [[119905 119878]] it holds that 120572(1199071) equiv
120572(1199072) on [[119905 119878]] Nonanticipative strategies for Player II on[119905 119904] 120573 U119905119904 rarr V119905119904 are defined similarly The set ofall nonanticipative strategies 120572 V119905119904 rarr U119905119904 for PlayerI on [119905 119904] is denoted by A119905119904 The set of all nonanticipativestrategies 120573 U119905119904 rarr V119905119904 for Player II on [119905 119904] is denoted byB119905119904 (Recall that [[119905 119878]] = (119903 120596) isin [0 119879] times Ω 119905 le 119903 le 119878(120596))
For given processes 119906(sdot) isin U119905119879 119907(sdot) isin V119905119879 initial data119905 isin [0 119879] 120574119905 isin Λ the cost functional is defined as follows
119869 (120574119905 119906 119907) = 119884120574119905119906119907
(119905) 120574119905 isin Λ (21)
where the process 119884120574119905119906119907 is defined by functional FBSDE (17)
For 120574119905 isin Λ the lower and the upper value functions of ourSDGs are defined as
119882(120574119905) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (22)
119880 (120574119905) = esssup120572isinA119905119879
essinf119907isinV119905119879
119869 (120574119905 120572 (119907) 119907) (23)
As we know the essential infimum and essential supre-mum on a family of random variables are still random vari-ables But by applying the method introduced by Buckdahnand Li [10] we get119882(120574119905) and 119880(120574119905) are deterministic
Proposition 12 For any 119905 isin [0 119879] 120574119905 isin Λ 119882(120574119905) is a deter-ministic function in the sense that119882(120574119905) = 119864[119882(120574119905)] 119875-as
Proof Let 119867 denote the Cameron-Martin space of all abso-lutely continuous elements ℎ isin Ω whose derivative ℎ belongsto 119871
2([0 119879]R119889
)For any ℎ isin 119867 we define the mapping 120591ℎ120596 = 120596 + ℎ 120596 isin
Ω It is easy to check that 120591ℎ Ω rarr Ω is a bijectionand its law is given by 119875 ∘ [120591ℎ]
minus1= expint119879
0ℎ(119904)119889119861(119904) minus
(12) int119879
0|ℎ(119904)|
2119889119904119875 For any fixed 119905 isin [0 119879] set 119867119905 = ℎ isin
119867 | ℎ(sdot) = ℎ(sdot and 119905) The proof can be separated into thefollowing four steps
(1) For all 119906 isin U119905119879 119907 isin V119905119879 ℎ isin 119867119905 119869(120574119905 119906 119907)(120591ℎ) =
119869(120574119905 119906(120591ℎ) 119907(120591ℎ)) 119875-asFirst we make the transformation for the functional SDE
as follows
119883120574119905119906119907
(119904)∘(120591ℎ)= (120574119905 (119905)+int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
+int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)
= 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
(24)
then from the uniqueness of the solution of the functionalSDE we get
119883120574119905119906119907
(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as(25)
Similarly using the transformation to the BSDE in (17) andcomparing the obtained equation with the BSDE obtainedfrom (17) by replacing the transformed control process119906(120591ℎ) 119907(120591ℎ) for 119906 119907 due to the uniqueness of the solution offunctional BSDE we obtain
119884120574119905119906119907
(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as
119885120574119905119906119907
(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)
119889119904119889119875-ae on [0 119879] times Ω
(26)
Hence
119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)
(2) For 120573 isin B119905119879 ℎ isin 119867119905 let 120573ℎ(119906) = 120573(119906(120591minusℎ))(120591ℎ) 119906 isin
U119905119879 Then 120573ℎisin B119905119879
Obviously 120573ℎ U119905119879 rarr V119905119879 And it is nonanticipating
In fact given an F-stopping time 119878 Ω rarr [119905 119879] and 1199061 1199062 isin
U119905119879 with 1199061 equiv 1199062 on [[119905 119878]] Accordingly 1199061(120591minusℎ) equiv 1199062(120591minusℎ)
on [[119905 119878(120591minusℎ)]] Thus
120573ℎ(1199061) = 120573 (1199061 (120591minusℎ)) (120591ℎ) = 120573 (1199062 (120591minusℎ)) (120591ℎ)
= 120573ℎ(1199062) on [[119905 119878]]
(28)
(3) For any ℎ isin 119867119905 and 120573 isin B119905119879 we have
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)
In fact for convenience setting 119868(120574119905 120573) = esssup119906isinU119905119879
119869(120574119905 119906
120573(119906)) 120573 isin B119905119879 we know 119868(120574119905 120573) ge 119869(120574119905 119906 120573(119906)) Then
6 Mathematical Problems in Engineering
119868(120574119905 120573)(120591ℎ) ge 119869(120574119905 119906 120573(119906))(120591ℎ) 119875-as for all 119906 isin U119905119879Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)
From the definition of essential supremum for anyrandom variable 120585 which satisfies 120585 ge 119869(120574119905 119906 120573(119906))(120591ℎ) wehave 120585(120591minusℎ) ge 119869(120574119905 119906 120573(119906)) 119875-as for all 119906 isin U119905119879 So120585(120591minusℎ) ge 119868(120574119905 120573) 119875-as that is 120585 ge 119868(120574119905 120573)(120591ℎ) 119875-as Thus
119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
119875-as for any 119906 isin U119905119879
(31)
Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)
From above we get
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)
(4) Under the Girsanov transformation 120591ℎ 119882(120574119905) isinvariant that is
119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)
In fact we can prove essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) =
essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) 119875-as for all ℎ isin 119867119905 which issimilar to the above step From the above three steps for allℎ isin 119867119905 we get
119882(120574119905) (120591ℎ) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906))
= 119882 (120574119905) 119875-as(35)
Note that 119906(120591ℎ) | 119906(sdot) isin U119905119879 = U119905119879 and 120573ℎ
| 120573 isin
B119905119879 = B119905119879 have been used in the above equalities So forany ℎ isin 119867119905119882(120574119905)(120591ℎ) = 119882(120574119905) 119875-as Thanks to 119882(120574119905) isF119905-measurable this relation holds true for all ℎ isin 119867
To finish the proof we also need the auxiliary lemma asfollows
Lemma 13 Let 120577 be a random variable defined over theclassical Wiener space (ΩF119879 119875) such that 120577(120591ℎ) = 120577 119875-asfor any ℎ isin 119867 Then 120577 = 119864120577 119875-as
Proof From Lemma 13 in Buckdahn and Li [10] we know forany 119860 isin B(R) 120593 isin 119871
2([0 119879]R119889
)
119864[1120577isin119860 expint
119879
0
120593 (119904) 119889119861 (119904)]
= 119864 [1120577isin119860] 119864 [expint
119879
0
120593 (119904) 119889119861 (119904)]
(36)
For any 120593 = sum119873
119894=11205931198941(119905
119894minus1119905119894] where 120593119894 isin R119889 for 0 le 119894 le 119873 and
119905119894119873
119894=0is a finite partition of [0 119879] from (36)
119864[1120577isin119860 exp(
119873
sum
119894=1
120593119894119861 (119905119894) minus 119861 (119905119894minus1))]
= 119864 [1120577isin119860] 119864[exp119873
sum
119894=1
120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]
(37)
Therefore for any nonnegative integer 119896119894
119864[1120577isin119860
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
= 119864 [1120577isin119860] 119864[
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
(38)
So for any polynomial function 119876 we have
119864 [1120577isin119860119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
= 119864 [1120577isin119860] 119864 [119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
(39)
Furthermore for any 119876 isin 119862119887(R119899) we still have (39)
Combining the arbitrariness of 119860 isin B(R) we obtain 120577 isindependent of (119861(1199051) minus 119861(1199050) 119861(119905119899) minus 119861(119905119899minus1)) for allpartition of [0 119879] Therefore 120577 is independent of F119879 whichimplies 120577 is independent of itself that is 119864120577 = 120577 119875-as
In Ji and Yang [14] they proved the following estimates
Mathematical Problems in Engineering 7
Lemma 14 Under the assumption (H) there exists someconstant 119862 gt 0 such that for any 119905 isin [0 119879] 120574119905 120574119905 isin Λ 119906(sdot) isin
U 119907(sdot) isin V
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905]le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119883
120574119905119906119907
(119904) minus 119883120574119905119906119907
(119904)10038161003816100381610038161003816
2
| F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
119864 [ sup119904isin[119905119879]
1003816100381610038161003816119884120574119905119906119907
(119904)10038161003816100381610038162+int
119879
119905
1003816100381610038161003816119885120574119905119906119907
(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119884120574119905119906119907
(119904) minus 119884120574119905119906119907
(119904)10038161003816100381610038161003816
2
+int
119879
119905
10038161003816100381610038161003816119885120574119905119906119907
(119904) minus 119885120574119905119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
(40)From the definition of119882(120574119905) and Lemma 14 we have the
following property
Lemma 15 There exists some constant 119862 gt 0 such that for all0 le 119905 le 119879 120574119905 120574119905 isin Λ
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817
(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171205741199051003817100381710038171003817)
(41)
Nowwe adopt Pengrsquos notion of stochastic backward semi-group (which was first introduced by Peng [9] to prove theDPP for stochastic control problems) to discuss a generalizedDPP for our SDG (17) (22) First we define the family ofbackward semigroups associated with FBSDE (17)
For given 119905 isin [0 119879] 120574119905 isin Λ a number 120575 isin (0 119879 minus 119905]admissible control processes 119906(sdot) isin U119905119905+120575 119907(sdot) isin V119905119905+120575 weset
119866120574119905119906119907
119904119905+120575[120578] =
120574119905119906119907
(119904) 119904 isin [119905 119905 + 120575] (42)
where 120578 isin 1198712(ΩF119905+120575
119905 119875R) (
120574119905119906119907
(119904) 119885120574119905119906119907
(119904))119905le119904le119905+120575
solves the following functional FBSDE on [119905 119905 + 120575]
119889120574119905119906119907
(119904)
= minus119891 (119904 119883120574119905119906119907
119904
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
120574119905119906119907
(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]
(43)Also we have
119866120574119905119906119907
119905119879[Φ (119883
120574119905119906119907
119879)] = 119866
120574119905119906119907
119905119905+120575[119884
120574119905119906119907
(119905 + 120575)] (44)Theorem 16 Suppose (H) holds true the lower value function119882(120574119905) satisfies the followingDPP for any 119905 isin [0 119879] 120574119905 isin Λ 120575 gt
0
119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (45)
The proof is given in the appendix
4 Viscosity Solutions of Path-DependentHJBI Equation
Now we study the following path-dependent PDEs
119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0
119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ
(46)
119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0
119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ
(47)
where
119867minus(120574119905119882119863119909119882119863119909119909119882)
= sup119906isin119880
inf119907isin119881
119867(120574119905119882119863119909119882119863119909119909119882119906 119907)
119867+(120574119905 119880119863119909119880119863119909119909119880)
= inf119907isin119881
sup119906isin119880
119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)
119867 (120574119905 119910 119901 119883 119906 119907)
=1
2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)
+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)
(48)
where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of
119889 times 119889 symmetric matrices)We will show that the value function119882(120574119905) (resp 119880(120574119905))
defined in (22) (resp (23)) is a viscosity solution of thecorresponding equation (46) (resp (47)) First we give thedefinition of viscosity solution for this kind of PDEs Formore information on viscosity solution the reader is referredto Crandall et al [22]
Definition 17 A real-valued Λ-continuous function 119882 isin
C(Λ) is called
(i) a viscosity subsolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ ge 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)
(ii) a viscosity supersolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ le 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)
(iii) a viscosity solution of (46) if it is both a viscosity sub-and supersolution of (46)
Theorem 18 Assume (H) holds the lower value function119882 isa viscosity solution of path-dependent HJBI Equation (46) onΛ the upper value function 119880 is a viscosity solution of (47)
8 Mathematical Problems in Engineering
First we prove some helpful lemmas For some fixed Γ isin
C12
119897119887(Λ) denote
119865 (120574119904 119910 119911 119906 119907)
= 119863119904Γ (120574119904) +1
2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))
+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)
+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)
(51)
where (120574119904 119910 119911 119906 119907) isin Λ timesR timesR119889times 119880 times 119881
Consider the following BSDE
minus 1198891198841119906119907
(119904)
= 119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198851119906119907
(119904) 119889119861 (119904)
1198841119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]
(52)
Lemma 19 For every 119904 isin [119905 119905 + 120575] one has the following
1198841119906119907
(119904) = 119866120574119905119906119907
119904119905+120575[Γ (119883
120574119905119906119907
119905+120575)] minus Γ (119883
120574119905119906119907
119904) 119875-as (53)
Proof Note119866120574119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] is defined as119866120574
119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] =
119884119906119907
(119904) through the following BSDE
minus 119889119884119906119907
(119904) = 119891 (119883120574119905119906119907
119904 119884
119906119907(119904) 119885
119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 119885119906119907
(119904) 119889119861 (119904)
119884119906119907
(119905 + 120575) = Γ (119883120574119905119906119907
119905+120575) 119904 isin [119905 119905 + 120575]
(54)
Using Itorsquos formula to Γ(119883120574119905119906119907
119904) we have
119889 (119884119906119907
(119904) minus Γ (119883120574119905119906119907
119904)) = 119889119884
1119906119907(119904) (55)
Combined with 119884119906119907
(119905 + 120575) minus Γ(119883120574119905119906119907
119905+120575) = 0 = 119884
1119906119907(119905 + 120575) we
get the desired result
Now consider the following BSDE
minus 1198891198842119906119907
(119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198852119906119907
(119904) 119889119861 (119904)
(56)
1198842119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)
Then we have the following lemma
Lemma 20 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816le 119862120575
32 119875-as (58)
where 119862 is independent of the control processes 119906 119907
Proof From Lemma 14 we know there exists some constant119862 gt 0 such that
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905] le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172) (59)
combined with
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]
le 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
10038161003816100381610038161003816100381610038161003816
2
| F119905]
+ 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)
10038161003816100381610038161003816100381610038161003816
2
| F119905]
(60)
we have
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)
From (52) and (54) using Lemma 8 set
1205851 = 1205852 = 0
119892 (119904 119910 119911) = 119865 (119883120574119905119906119907
119904 119910 119911 119906119904 119907119904)
1205931 (119904) = 0
1205932 (119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906119904 119907119904)
minus 119865 (119883120574119905119906119907
119904 119884
2119906119907(119904) 119885
2119906119907(119904) 119906119904 119907119904)
(62)
Denote 1205880(119903) = (1 + |119909|2)119903 due to 119887 120590 119891 are Lipishtiz and
that they are of linear growth Γ isin C12
119887 |1205932(119904)| le 1205880(|119883
120574119905119906119907
119904minus
120574119905(119905)|) we have
119864[int
119905+120575
119905
(100381610038161003816100381610038161198841119906119907
(119904) minus1198842119906119907
(119904)10038161003816100381610038161003816
2
+100381610038161003816100381610038161198851119906119907
(119904) minus1198852119906119907
(119904)10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864[int
119905+120575
119905
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) 119889119904 | F119905]
le 119862120575119864[ sup119904isin[119905119905+120575]
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) | F119905]
le 1198621205752
(63)
Mathematical Problems in Engineering 9
So100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816
=10038161003816100381610038161003816119864 [(119884
1119906119907(119904) minus 119884
2119906119907(119904)) | F119905]
10038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
119864 [int
119905+120575
119905
(119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904))
minus119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]
100381610038161003816100381610038161003816100381610038161003816
le 119862119864[int
119905+120575
119905
[1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
+100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816] 119889119904 | F119905]
le 119862119864[int
119905+120575
119905
1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) 119889119904 | F119905]
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
le 11986212057532
(64)
Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation
minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]
1198840 (119905 + 120575) = 0
(65)
where 1198650(120574119905 119910 119911) = sup119906isin119880
inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as
1198840 (119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) (66)
Proof First we define a function as follows
1198651 (120574119905 119910 119911 119906) = inf119907isin119881
119865 (120574119905 119910 119911 119906 119907)
(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880
(67)
Consider the following equation
minus1198891198843119906
(119904) = 1198651 (120574119905 1198843119906
(119904) 1198853119906
(119904) 119906 (119904)) 119889119904
minus1198853119906
(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]
1198843119906
(119905 + 120575) = 0
(68)
according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906
1198853119906
) solving (68) Also
1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as for every 119906 isin U119905119905+120575
(69)
In fact according to the definition of 1198651 and Lemma 7 wehave
1198843119906
(119905) le 1198842119906119907
(119905) 119875-as for any 119907 isin V119905119905+120575
for every 119906 isin U119905119905+120575
(70)
On the other side we have the existence of a measurablefunction 119907
4 R timesR119889
times 119880 rarr 119881 such that
1198651 (120574119905 119910 119911 119906)
= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906
(71)
Set 1199074(119904) = 1199074(119884
3119906(119904) 119885
3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know
1199074isin V119905119905+120575 Therefore (1198843119906
1198853119906
) = (11988421199061199074
11988521199061199074
) which isdue to the uniqueness of the solution of (68) In particular1198843119906
(119905) = 11988421199061199074
(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-119886119904 for every 119906 isin U119905119905+120575
Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880
1198651(120574119905 119910 119911 119906) wealso derive
1198840 (119905) = esssup119906isinU119905119905+120575
1198843119906
(119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as
(72)
Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575
32 119875-as
(73)
where the constant 119862 is independent of the control processes119906 119907
Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as
100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
2
le 119862120575
119864 [int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904] le 119862120575
(74)
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
[14] investigated a controlled system governed by a functionalforward-backward stochastic differential equation (FBSDE)and proved the value function to be the viscosity solution ofthe related path-dependent HJB equation
In this paper inspired by [10 14] we will investigatethe SDG problems of the functional FBSDEs Preciselythe dynamics of our SDGs is described by the followingfunctional SDE
119889119883120574119905119906119907
(119904) = 119887 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119904
+ 120590 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]
119883120574119905119906119907
119905= 120574119905
(1)
And the cost functional 119869(120574119905 119906 119907) is defined as119884120574119905119906119907
(119905)whichis the solution of the following functional BSDE
119889119884120574119905119906119907
(119904) = minus 119891 (119883120574119905119906119907
119904 119884
120574119905119906119907
(119904) 119885120574119905119906119907
(119904)
119906 (119904) 119907 (119904)) 119889119904 + 119885120574119905119906119907
(119904) 119889119861 (119904)
119884120574119905119906119907
(119879) = Φ (119883120574119905119906119907
119879) 119904 isin [119905 119879]
(2)
where 120574119905 is a path on [0 119905] The driver 119891 and Φ canbe interpreted as the running cost and the terminal costrespectively Also they depend on the past history of thedynamics Equations (1) and (2) compose a decoupled func-tional FBSDE The concrete conditions on 119887 120590 119891 Φ areshown in the later section
In the context we adopt the strategy against control formThe cost functional 119869(120574119905 119906 119907) is explained as a payoff forplayer I and as a cost for player II The aim of this paper isto show the following lower and upper value functions
119882(120574119905) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906))
119880 (120574119905) = esssup120572isinA119905119879
essinf119907isinV119905119879
119869 (120574119905 120572 (119907) 119907)
(3)
are the viscosity solutions of the following path-dependentHJBI equations respectively
119863119905119882(120574119905) + sup119906isin119880
inf119907isin119881
119867(120574119905119882119863119909119882119863119909119909119882119906 119907) = 0
119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ
119863119905119880(120574119905) + inf119907isin119881
sup119906isin119880
119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907) = 0
119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ
(4)
where
119867(120574119905 119910 119901 119883 119906 119907) =1
2tr (120590120590119879 (120574119905 119906 119907)119883)
+ 119901 sdot 119887 (120574119905 119906 119907)
+ 119891 (120574119905 119910 119901 sdot 120590 (120574119905 119906 119907) 119906 119907)
(5)
where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of
119889 times 119889 symmetric matrices)To solve the above SDG we need the functional Itorsquos
calculus and path-dependent PDEs which are recently intro-duced by Dupire [15] (for a recent account of this theorythe reader may consult [16ndash18] And under the framework offunctional Itorsquos calculus for the non-Markovian BSDEs PengandWang [19] derived a nonlinear Feynman-Kac formula forclassical solutions of path-dependent PDEs For the furtherdevelopment the readers may refer to [20 21])
In this paper we apply the Girsanov transformationmethod in Buckdahn and Li [10] to prove the determinacyof the value functions which is different from the methodintroduced by Peng [7 9] Making use of this methodand the functional Itorsquos calculus (introduced by Dupire [15]and developed by Cont and Fournie [16ndash18]) we completethe study of the zero-sum two-player SDGs in the non-Markovian case and present the lower and upper valuefunctions of our SDG are the viscosity solutions of thecorresponding path-dependent HJBI equations respectively
Different from the HJBI equations developed for stochas-tic delay systems we establish the DPP and derive the HJBIequation in the new framework of functional Itorsquos calculus
This paper is organized as follows Section 2 recalls thefunctional Itorsquos calculus and the well-known results of BSDEswe will use later In Section 3 we formulate our SDGs andget the corresponding DPP Based on the obtained DPP inSection 4 we derive the main result of the paper the lowerand upper value functions are the viscosity solutions of theassociated path-dependent HJBI equations respectively Andwe add the proof for the DPP in the appendix
2 Preliminaries
21 Functional Itorsquos Calculus We present some preliminariesfor functional Itorsquos calculus introduced firstly by Dupire [15]Here we follow the notations in [15]
Let 119879 gt 0 be fixed For each 119905 isin [0 119879] we denote Λ 119905 theset of cadlag functions from [0 119905] to R119889
For 120574 isin Λ 119879 denote 120574(119904) by the value of 120574 at time 119904 isin [0 119879]Thus 120574 = (120574(119904))0le119904le119879 is a cadlag process on [0 119879] and its valueat time 119904 is 120574(119904) 120574119905 = (120574(119904))0le119904le119905 isin Λ 119905 is the path of 120574 upto time 119905 We denote Λ = ⋃
119905isin[0119879]Λ 119905 For each 120574119905 isin Λ and
119909 isin R119889 120574119905(119904) is denoted by the value of 120574119905 at 119904 isin [0 119905] and120574119909
119905= (120574119905(119904)0le119904lt119905 120574119905(119905) + 119909) which is also an element in Λ 119905We now introduce a distance onΛ Let ⟨sdot sdot⟩ and | sdot| denote
the inner product and norm inR119889 For each 0 le 119905 119905 le 119879 and120574119905 120574119905 isin Λ we set
10038171003817100381710038171205741199051003817100381710038171003817 = sup
119904isin[0119905]
1003816100381610038161003816120574119905 (119904)1003816100381610038161003816
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 = sup119904isin[0119905or119905]
1003816100381610038161003816120574119905 (119904 and 119905) minus 120574119905(119904 and 119905)
1003816100381610038161003816
119889infin (120574119905 120574119905) = sup119904isin[0119905or119905]
1003816100381610038161003816120574119905 (119904 and 119905) minus 120574119905(119904 and 119905)
1003816100381610038161003816 +1003816100381610038161003816119905 minus 119905
1003816100381610038161003816
(6)
It is obvious that Λ 119905 is a Banach space with respect to sdot Since Λ is not a linear space 119889infin is not a norm
Mathematical Problems in Engineering 3
Definition 1 A functional 119906 Λ 997891rarr R is Λ-continuous at120574119905 isin Λ if for any 120576 gt 0 there exists 120575 gt 0 such that for each120574119905isin Λ with 119889infin(120574119905 120574119905) lt 120575 one has |119906(120574119905) minus 119906(120574
119905)| lt 120576
119906 is said to be Λ-continuous if it is Λ-continuous at each120574119905 isin Λ
Definition 2 Let 119907 Λ 997891rarr R and 120574119905 isin Λ be given If thereexists 119901 isin R119889 such that
119907 (120574119909
119905) = 119907 (120574119905) + ⟨119901 119909⟩ + 119900 (|119909|) as 119909 997888rarr 0 119909 isin R
119889
(7)
Then we say that 119907 is (vertically) differentiable at 119905 and denotethe gradient of 119863119909119907(120574119905) = 119901 sdot 119907 is said to be verticallydifferentiable in Λ if 119863119909119907(120574119905) exists for each 120574119905 isin Λ TheHessian119863119909119909119907(120574119905) can be defined similarly It is an 119878(119889)-valuedfunction defined on Λ where 119878(119889) is the space of all 119889 times 119889
symmetric matrices
For each 120574119905 isin Λ we denote 120574119905119904(119903) = 120574119905(119903)1[0119905)(119903) +
120574119905(119905)1[119905119904](119903) 119903 isin [0 119904] It is clear that 120574119905119904 isin Λ 119904
Definition 3 For a given 120574119905 isin Λ if one has
119907 (120574119905119904) = 119907 (120574119905) + 119886 (119904 minus 119905) + 119900 (|119904 minus 119905|) as 119904 997888rarr 119905 119904 ge 119905
(8)
then we say that 119907(120574119905) is (horizontally) differentiable in 119905
at 120574119905 and denote 119863119905119907(120574119905) = 119886 119907 is said to be horizontallydifferentiable in Λ if119863119905119907(120574119905) exists for each 120574119905 isin Λ
Definition 4 Define C119895119896(Λ) as the set of function 119907 =
(119907(120574119905))120574119905isinΛ defined on Λ which are 119895 times horizontally and
119896 times vertically differentiable in Λ such that all thesederivatives are Λ-continuous
The following is about the functional Itorsquos formula whichwas firstly obtained by Dupire [15] and then developed byCont and Fournie [18] for a more general formulation
Theorem 5 (functional Itorsquos formula) Let (ΩF (F119905)119905isin[0119879]
119875) be a probability space if 119883 is a continuous semimartingaleand 119907 is in C12
(Λ) then for any 119905 isin [0 119879)
119907 (119883119905) minus 119907 (1198830) = int
119905
0
119863119904119907 (119883119904) 119889119904 + int
119905
0
119863119909119907 (119883119904) 119889119883 (119904)
+1
2int
119905
0
119863119909119909119907 (119883119904) 119889 ⟨119883⟩ (119904) 119875-as
(9)
22 BSDEs In this section we recollect some importantresults which will be used in our SDG problems
Let (ΩF 119875) be the Wiener space where Ω is the set ofcontinuous functions from [0 119879] to R119889 starting from 0 (Ω =
1198620([0 119879]R119889)) 119879 is an arbitrarily fixed real time horizonF
is the completed Borel 120590-algebra overΩ and 119875 is the Wienermeasure Let 119861 be the canonical process 119861(120596 119904) = 120596119904 119904 isin
[0 119879] 120596 isin Ω We denote by F = F119904 0 le 119904 le 119879 the natural
filtration generated by 119861(119905)119905ge0 and augmented by all 119875-nullsets that is F119904 = 120590119861(119903) 119903 le 119904 or N119875 119904 isin [0 119879] F119904
119905=
120590(119861(119903)minus119861(119905) 119905 le 119903 le 119904)orN119875 whereN119875 is the set of all119875-nullsubsets First we present two spaces of processes as follows
S2(0 119879R
119899) = (120595(119905))
0le119905le119879R
119899-valued
F-adapted continuous process
119864 [ sup0le119905le119879
1003816100381610038161003816120595 (119905)10038161003816100381610038162] lt +infin
H2(0 119879R
119899) = (120595(119905))
0le119905le119879
R119899-valued
F-progressively measurable process
100381710038171003817100381712059510038171003817100381710038172= 119864[int
119879
0
1003816100381610038161003816120595 (119905)10038161003816100381610038162119889119905] lt +infin
(10)
Consider 119892 Ω times [0 119879] times R times R119889rarr R for every (119910 119911)
in R times R119889 (119892(119905 119910 119911))119905isin[0119879] is progressively measurable andsatisfies the following conditions
(A1) there exists a constant 119862 ge 0 such that 119875-as for all119905 isin [0 119879] 119910
1 1199102 isin R 1199111 1199112 isin R119889
1003816100381610038161003816119892 (119905 1199101 1199111) minus 119892 (119905 1199102 1199112)1003816100381610038161003816 le 119862 (
10038161003816100381610038161199101 minus 11991021003816100381610038161003816 +
10038161003816100381610038161199111 minus 11991121003816100381610038161003816) (11)
(A2) 119892(sdot 0 0) isin H2(0 119879R)
In the following we suppose the driver 119892 of a BSDEsatisfies (A1) and (A2)
Lemma 6 Under the assumptions (A1) and (A2) for anyrandom variable 120585 isin 119871
2(ΩF119879 119875R) the BSDE
119910 (119905) = 120585 + int
119879
119905
119892 (119904 119910 (119904) 119911 (119904)) 119889119904 minus int
119879
119905
119911 (119904) 119889119861 (119904)
0 le 119905 le 119879
(12)
has a unique adapted solution
(119910(119905) 119911(119905))119905isin[0119879]
isin S2(0 119879R) timesH
2(0 119879R
119889) (13)
The readers may refer to Pardoux and Peng [1] forthe above well-known existence and uniqueness results onBSDEs
Lemma 7 (comparison theorem) Given two coefficients 11989211198922 satisfying (A1) (A2) and two terminal values 1205851 1205852 isin
1198712(ΩF119879 119875R) by (119910
1 119911
1) and (119910
2 119911
2) one denotes the
solution of a BSDE with the data (1205851 1198921) and (1205852 1198922)respectively Then one has the following
(i) If 1205851 ge 1205852 and 1198921 ge 1198922 P-as then 1199101(119905) ge 119910
2(119905)
P-as for all 119905 isin [0 119879]
4 Mathematical Problems in Engineering
(ii) (Strict monotonicity) If in addition to (i) one alsoassumes that 119875(1205851 gt 1205852) gt 0 then 119875(119910
1(119905) gt 119910
2(119905)) gt
0 0 le 119905 le 119879 and in particular 1199101(0) gt 119910
2(0)
With the notations in Lemma 7 we assume that for some119892 Ω times [0 119879] times R times R119889
rarr R satisfying (A1) and (A2) thedrivers 119892119894 have the following form
119892119894 (119904 119910119894(119904) 119911
119894(119904)) = 119892 (119904 119910
119894(119904) 119911
119894(119904)) + 120593119894 (119904)
119889119904119889119875-ae 119894 = 1 2
(14)
where 120593119894 isin H2(0 119879R) Then the following results hold true
for all terminal values 1205851 1205852 isin 1198712(ΩF119879 119875R)
Lemma 8 The difference of the solutions (1199101 119911
1) and (119910
2 119911
2)
of BSDE (12) with the data (1205851 1198921) and (1205852 1198922) respectivelysatisfies the following estimate
100381610038161003816100381610038161199101(119905)minus119910
2(119905)
10038161003816100381610038161003816
2
+1
2119864 [int
119879
119905
119890120573(119904minus119905)
(100381610038161003816100381610038161199101(119904)minus119910
2(119904)
10038161003816100381610038161003816
2
+100381610038161003816100381610038161199111(119904) minus 119911
2(119904)
10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864 [119890120573(119879minus119905)10038161003816100381610038161205851 minus 1205852
10038161003816100381610038162| F119905]
+ 119864 [int
119879
119905
119890120573(119904minus119905)10038161003816100381610038161205931 (119904) minus 1205932 (119904)
10038161003816100381610038162119889119904 | F119905]
P-as forall0 le 119905 le 119879 where 120573 = 16 (1 + 1198622)
(15)
Proof The reader may refer to Proposition 21 in El Karoui etal [5] or Theorem 23 in Peng [9] for the details
3 A DPP for Stochastic Differential Games ofFunctional FBSDEs
In this section we consider the SDGs of functional FBSDEsFirst we introduce the background of SDGs Suppose
that the control state spaces 119880119881 are compact metric spacesU (resp V) is the control set of all 119880 (resp 119881)-valuedF-progressively measurable processes for the first (respsecond) player If 119906 isin U (resp 119907 isin V) we call 119906 (resp 119907)an admissible control
Let us give the following mappings
119887 Λ times 119880 times 119881 997888rarr R119899
120590 Λ times 119880 times 119881 997888rarr R119899times119889
119891 Λ timesR timesR119889times 119880 times 119881 997888rarr R
(16)
For given admissible controls 119906(sdot) isin U 119907(sdot) isin V and 119905 isin
[0 119879] 120574119905 isin Λ we consider the following functional forward-backward stochastic system
119889119883120574119905119906119907
(119904) = 119887 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119904
+ 120590 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]
119889119884120574119905119906119907
(119904) = minus119891 (119883120574119905119906119907
119904 119884
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
119883120574119905119906119907
119905= 120574119905
119884120574119905119906119907
(119879) = Φ (119883120574119905119906119907
119879)
(17)
(H) (i) For all 119905 isin [0 119879] 119906 isin 119880 119907 isin 119881 119909119905 isin Λ 119910 isin
R 119911 isin R119889 119887(119909119905 119906 119907) 120590(119909119905 119906 119907) and 119891(119909119905 119910 119911 119906 119907)
areF119905-measurable(ii) There exists a constant 119862 gt 0 such that for all 119905 isin
[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091
119905 119909
2
119905isin Λ
10038161003816100381610038161003816119887 (119909
1
119905 119906 119907) minus 119887 (119909
2
119905 119906 119907)
10038161003816100381610038161003816+10038161003816100381610038161003816120590 (119909
1
119905 119906 119907) minus 120590 (119909
2
119905 119906 119907)
10038161003816100381610038161003816
le 119862100381710038171003817100381710038171199091
119905minus 119909
2
119905
10038171003817100381710038171003817
(18)1003816100381610038161003816119887 (119909119905 119906 119907)
1003816100381610038161003816 +1003816100381610038161003816120590 (119909119905 119906 119907)
1003816100381610038161003816 le 119862 (1 +1003817100381710038171003817119909119905
1003817100381710038171003817) for any 119909119905 isin Λ
(19)
(iii) There exists a constant 119862 gt 0 such that for all 119905 isin
[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091
119905 119909
2
119905isin Λ 119910
1 119910
2isin
R 1199111 119911
2isin R119889
10038161003816100381610038161003816119891 (119909
1
119905 119910
1 119911
1 119906 119907) minus 119891 (119909
2
119905 119910
2 119911
2 119906 119907)
10038161003816100381610038161003816
le 119862 (100381710038171003817100381710038171199091
119905minus 119909
2
119905
10038171003817100381710038171003817+100381610038161003816100381610038161199101minus 119910
210038161003816100381610038161003816+100381610038161003816100381610038161199111minus 119911
210038161003816100381610038161003816)
10038161003816100381610038161003816Φ (119909
1
119879) minus Φ (119909
2
119879)10038161003816100381610038161003816le 119862
100381710038171003817100381710038171199091
119879minus 119909
2
119879
10038171003817100381710038171003817
1003816100381610038161003816119891 (119909119905 0 0 119906 119907)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171199091199051003817100381710038171003817)
1003816100381610038161003816Φ (119909119879)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171199091198791003817100381710038171003817) for any 119909119905 isin Λ
(20)
Theorem 9 Under the assumption (H) there exists a uniquesolution (119883 119884 119885) isin S2
(0 119879R119899) timesS2
(0 119879R) timesH2(0 119879R119889
)
solving (17)
We recall the subspaces of admissible controls and thedefinitions of admissible strategies as follows which aresimilar to [10]
Definition 10 An admissible control process 119906 = (119906119903)119903isin[119905119904]
(resp 119907 = (119907119903)119903isin[119905119904]) for Player I (resp II) on [119905 119904] is anF119903-progressively measurable 119880 (resp 119881)-valued process Theset of all admissible controls for Player I (resp II) on [119905 119904]
is denoted byU119905119904 (respV119905119904) If 119875119906 equiv 119906 119886119890 119894119899 [119905 119904] = 1one will identify both processes 119906 and 119906 inU119905119904 Similarly oneinterprets 119907 equiv 119907 on [119905 119904] inV119905119904
Mathematical Problems in Engineering 5
Definition 11 A nonanticipative strategy for Player I on[119905 119904] (119905 lt 119904 le 119879) is a mapping 120572 V119905119904 rarr U119905119904 suchthat for any F-stopping time 119878 Ω rarr [119905 119904] and any1199071 1199072 isin V119905119904 with 1199071 equiv 1199072 on [[119905 119878]] it holds that 120572(1199071) equiv
120572(1199072) on [[119905 119878]] Nonanticipative strategies for Player II on[119905 119904] 120573 U119905119904 rarr V119905119904 are defined similarly The set ofall nonanticipative strategies 120572 V119905119904 rarr U119905119904 for PlayerI on [119905 119904] is denoted by A119905119904 The set of all nonanticipativestrategies 120573 U119905119904 rarr V119905119904 for Player II on [119905 119904] is denoted byB119905119904 (Recall that [[119905 119878]] = (119903 120596) isin [0 119879] times Ω 119905 le 119903 le 119878(120596))
For given processes 119906(sdot) isin U119905119879 119907(sdot) isin V119905119879 initial data119905 isin [0 119879] 120574119905 isin Λ the cost functional is defined as follows
119869 (120574119905 119906 119907) = 119884120574119905119906119907
(119905) 120574119905 isin Λ (21)
where the process 119884120574119905119906119907 is defined by functional FBSDE (17)
For 120574119905 isin Λ the lower and the upper value functions of ourSDGs are defined as
119882(120574119905) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (22)
119880 (120574119905) = esssup120572isinA119905119879
essinf119907isinV119905119879
119869 (120574119905 120572 (119907) 119907) (23)
As we know the essential infimum and essential supre-mum on a family of random variables are still random vari-ables But by applying the method introduced by Buckdahnand Li [10] we get119882(120574119905) and 119880(120574119905) are deterministic
Proposition 12 For any 119905 isin [0 119879] 120574119905 isin Λ 119882(120574119905) is a deter-ministic function in the sense that119882(120574119905) = 119864[119882(120574119905)] 119875-as
Proof Let 119867 denote the Cameron-Martin space of all abso-lutely continuous elements ℎ isin Ω whose derivative ℎ belongsto 119871
2([0 119879]R119889
)For any ℎ isin 119867 we define the mapping 120591ℎ120596 = 120596 + ℎ 120596 isin
Ω It is easy to check that 120591ℎ Ω rarr Ω is a bijectionand its law is given by 119875 ∘ [120591ℎ]
minus1= expint119879
0ℎ(119904)119889119861(119904) minus
(12) int119879
0|ℎ(119904)|
2119889119904119875 For any fixed 119905 isin [0 119879] set 119867119905 = ℎ isin
119867 | ℎ(sdot) = ℎ(sdot and 119905) The proof can be separated into thefollowing four steps
(1) For all 119906 isin U119905119879 119907 isin V119905119879 ℎ isin 119867119905 119869(120574119905 119906 119907)(120591ℎ) =
119869(120574119905 119906(120591ℎ) 119907(120591ℎ)) 119875-asFirst we make the transformation for the functional SDE
as follows
119883120574119905119906119907
(119904)∘(120591ℎ)= (120574119905 (119905)+int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
+int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)
= 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
(24)
then from the uniqueness of the solution of the functionalSDE we get
119883120574119905119906119907
(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as(25)
Similarly using the transformation to the BSDE in (17) andcomparing the obtained equation with the BSDE obtainedfrom (17) by replacing the transformed control process119906(120591ℎ) 119907(120591ℎ) for 119906 119907 due to the uniqueness of the solution offunctional BSDE we obtain
119884120574119905119906119907
(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as
119885120574119905119906119907
(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)
119889119904119889119875-ae on [0 119879] times Ω
(26)
Hence
119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)
(2) For 120573 isin B119905119879 ℎ isin 119867119905 let 120573ℎ(119906) = 120573(119906(120591minusℎ))(120591ℎ) 119906 isin
U119905119879 Then 120573ℎisin B119905119879
Obviously 120573ℎ U119905119879 rarr V119905119879 And it is nonanticipating
In fact given an F-stopping time 119878 Ω rarr [119905 119879] and 1199061 1199062 isin
U119905119879 with 1199061 equiv 1199062 on [[119905 119878]] Accordingly 1199061(120591minusℎ) equiv 1199062(120591minusℎ)
on [[119905 119878(120591minusℎ)]] Thus
120573ℎ(1199061) = 120573 (1199061 (120591minusℎ)) (120591ℎ) = 120573 (1199062 (120591minusℎ)) (120591ℎ)
= 120573ℎ(1199062) on [[119905 119878]]
(28)
(3) For any ℎ isin 119867119905 and 120573 isin B119905119879 we have
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)
In fact for convenience setting 119868(120574119905 120573) = esssup119906isinU119905119879
119869(120574119905 119906
120573(119906)) 120573 isin B119905119879 we know 119868(120574119905 120573) ge 119869(120574119905 119906 120573(119906)) Then
6 Mathematical Problems in Engineering
119868(120574119905 120573)(120591ℎ) ge 119869(120574119905 119906 120573(119906))(120591ℎ) 119875-as for all 119906 isin U119905119879Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)
From the definition of essential supremum for anyrandom variable 120585 which satisfies 120585 ge 119869(120574119905 119906 120573(119906))(120591ℎ) wehave 120585(120591minusℎ) ge 119869(120574119905 119906 120573(119906)) 119875-as for all 119906 isin U119905119879 So120585(120591minusℎ) ge 119868(120574119905 120573) 119875-as that is 120585 ge 119868(120574119905 120573)(120591ℎ) 119875-as Thus
119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
119875-as for any 119906 isin U119905119879
(31)
Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)
From above we get
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)
(4) Under the Girsanov transformation 120591ℎ 119882(120574119905) isinvariant that is
119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)
In fact we can prove essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) =
essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) 119875-as for all ℎ isin 119867119905 which issimilar to the above step From the above three steps for allℎ isin 119867119905 we get
119882(120574119905) (120591ℎ) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906))
= 119882 (120574119905) 119875-as(35)
Note that 119906(120591ℎ) | 119906(sdot) isin U119905119879 = U119905119879 and 120573ℎ
| 120573 isin
B119905119879 = B119905119879 have been used in the above equalities So forany ℎ isin 119867119905119882(120574119905)(120591ℎ) = 119882(120574119905) 119875-as Thanks to 119882(120574119905) isF119905-measurable this relation holds true for all ℎ isin 119867
To finish the proof we also need the auxiliary lemma asfollows
Lemma 13 Let 120577 be a random variable defined over theclassical Wiener space (ΩF119879 119875) such that 120577(120591ℎ) = 120577 119875-asfor any ℎ isin 119867 Then 120577 = 119864120577 119875-as
Proof From Lemma 13 in Buckdahn and Li [10] we know forany 119860 isin B(R) 120593 isin 119871
2([0 119879]R119889
)
119864[1120577isin119860 expint
119879
0
120593 (119904) 119889119861 (119904)]
= 119864 [1120577isin119860] 119864 [expint
119879
0
120593 (119904) 119889119861 (119904)]
(36)
For any 120593 = sum119873
119894=11205931198941(119905
119894minus1119905119894] where 120593119894 isin R119889 for 0 le 119894 le 119873 and
119905119894119873
119894=0is a finite partition of [0 119879] from (36)
119864[1120577isin119860 exp(
119873
sum
119894=1
120593119894119861 (119905119894) minus 119861 (119905119894minus1))]
= 119864 [1120577isin119860] 119864[exp119873
sum
119894=1
120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]
(37)
Therefore for any nonnegative integer 119896119894
119864[1120577isin119860
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
= 119864 [1120577isin119860] 119864[
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
(38)
So for any polynomial function 119876 we have
119864 [1120577isin119860119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
= 119864 [1120577isin119860] 119864 [119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
(39)
Furthermore for any 119876 isin 119862119887(R119899) we still have (39)
Combining the arbitrariness of 119860 isin B(R) we obtain 120577 isindependent of (119861(1199051) minus 119861(1199050) 119861(119905119899) minus 119861(119905119899minus1)) for allpartition of [0 119879] Therefore 120577 is independent of F119879 whichimplies 120577 is independent of itself that is 119864120577 = 120577 119875-as
In Ji and Yang [14] they proved the following estimates
Mathematical Problems in Engineering 7
Lemma 14 Under the assumption (H) there exists someconstant 119862 gt 0 such that for any 119905 isin [0 119879] 120574119905 120574119905 isin Λ 119906(sdot) isin
U 119907(sdot) isin V
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905]le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119883
120574119905119906119907
(119904) minus 119883120574119905119906119907
(119904)10038161003816100381610038161003816
2
| F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
119864 [ sup119904isin[119905119879]
1003816100381610038161003816119884120574119905119906119907
(119904)10038161003816100381610038162+int
119879
119905
1003816100381610038161003816119885120574119905119906119907
(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119884120574119905119906119907
(119904) minus 119884120574119905119906119907
(119904)10038161003816100381610038161003816
2
+int
119879
119905
10038161003816100381610038161003816119885120574119905119906119907
(119904) minus 119885120574119905119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
(40)From the definition of119882(120574119905) and Lemma 14 we have the
following property
Lemma 15 There exists some constant 119862 gt 0 such that for all0 le 119905 le 119879 120574119905 120574119905 isin Λ
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817
(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171205741199051003817100381710038171003817)
(41)
Nowwe adopt Pengrsquos notion of stochastic backward semi-group (which was first introduced by Peng [9] to prove theDPP for stochastic control problems) to discuss a generalizedDPP for our SDG (17) (22) First we define the family ofbackward semigroups associated with FBSDE (17)
For given 119905 isin [0 119879] 120574119905 isin Λ a number 120575 isin (0 119879 minus 119905]admissible control processes 119906(sdot) isin U119905119905+120575 119907(sdot) isin V119905119905+120575 weset
119866120574119905119906119907
119904119905+120575[120578] =
120574119905119906119907
(119904) 119904 isin [119905 119905 + 120575] (42)
where 120578 isin 1198712(ΩF119905+120575
119905 119875R) (
120574119905119906119907
(119904) 119885120574119905119906119907
(119904))119905le119904le119905+120575
solves the following functional FBSDE on [119905 119905 + 120575]
119889120574119905119906119907
(119904)
= minus119891 (119904 119883120574119905119906119907
119904
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
120574119905119906119907
(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]
(43)Also we have
119866120574119905119906119907
119905119879[Φ (119883
120574119905119906119907
119879)] = 119866
120574119905119906119907
119905119905+120575[119884
120574119905119906119907
(119905 + 120575)] (44)Theorem 16 Suppose (H) holds true the lower value function119882(120574119905) satisfies the followingDPP for any 119905 isin [0 119879] 120574119905 isin Λ 120575 gt
0
119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (45)
The proof is given in the appendix
4 Viscosity Solutions of Path-DependentHJBI Equation
Now we study the following path-dependent PDEs
119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0
119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ
(46)
119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0
119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ
(47)
where
119867minus(120574119905119882119863119909119882119863119909119909119882)
= sup119906isin119880
inf119907isin119881
119867(120574119905119882119863119909119882119863119909119909119882119906 119907)
119867+(120574119905 119880119863119909119880119863119909119909119880)
= inf119907isin119881
sup119906isin119880
119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)
119867 (120574119905 119910 119901 119883 119906 119907)
=1
2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)
+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)
(48)
where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of
119889 times 119889 symmetric matrices)We will show that the value function119882(120574119905) (resp 119880(120574119905))
defined in (22) (resp (23)) is a viscosity solution of thecorresponding equation (46) (resp (47)) First we give thedefinition of viscosity solution for this kind of PDEs Formore information on viscosity solution the reader is referredto Crandall et al [22]
Definition 17 A real-valued Λ-continuous function 119882 isin
C(Λ) is called
(i) a viscosity subsolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ ge 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)
(ii) a viscosity supersolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ le 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)
(iii) a viscosity solution of (46) if it is both a viscosity sub-and supersolution of (46)
Theorem 18 Assume (H) holds the lower value function119882 isa viscosity solution of path-dependent HJBI Equation (46) onΛ the upper value function 119880 is a viscosity solution of (47)
8 Mathematical Problems in Engineering
First we prove some helpful lemmas For some fixed Γ isin
C12
119897119887(Λ) denote
119865 (120574119904 119910 119911 119906 119907)
= 119863119904Γ (120574119904) +1
2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))
+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)
+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)
(51)
where (120574119904 119910 119911 119906 119907) isin Λ timesR timesR119889times 119880 times 119881
Consider the following BSDE
minus 1198891198841119906119907
(119904)
= 119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198851119906119907
(119904) 119889119861 (119904)
1198841119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]
(52)
Lemma 19 For every 119904 isin [119905 119905 + 120575] one has the following
1198841119906119907
(119904) = 119866120574119905119906119907
119904119905+120575[Γ (119883
120574119905119906119907
119905+120575)] minus Γ (119883
120574119905119906119907
119904) 119875-as (53)
Proof Note119866120574119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] is defined as119866120574
119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] =
119884119906119907
(119904) through the following BSDE
minus 119889119884119906119907
(119904) = 119891 (119883120574119905119906119907
119904 119884
119906119907(119904) 119885
119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 119885119906119907
(119904) 119889119861 (119904)
119884119906119907
(119905 + 120575) = Γ (119883120574119905119906119907
119905+120575) 119904 isin [119905 119905 + 120575]
(54)
Using Itorsquos formula to Γ(119883120574119905119906119907
119904) we have
119889 (119884119906119907
(119904) minus Γ (119883120574119905119906119907
119904)) = 119889119884
1119906119907(119904) (55)
Combined with 119884119906119907
(119905 + 120575) minus Γ(119883120574119905119906119907
119905+120575) = 0 = 119884
1119906119907(119905 + 120575) we
get the desired result
Now consider the following BSDE
minus 1198891198842119906119907
(119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198852119906119907
(119904) 119889119861 (119904)
(56)
1198842119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)
Then we have the following lemma
Lemma 20 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816le 119862120575
32 119875-as (58)
where 119862 is independent of the control processes 119906 119907
Proof From Lemma 14 we know there exists some constant119862 gt 0 such that
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905] le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172) (59)
combined with
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]
le 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
10038161003816100381610038161003816100381610038161003816
2
| F119905]
+ 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)
10038161003816100381610038161003816100381610038161003816
2
| F119905]
(60)
we have
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)
From (52) and (54) using Lemma 8 set
1205851 = 1205852 = 0
119892 (119904 119910 119911) = 119865 (119883120574119905119906119907
119904 119910 119911 119906119904 119907119904)
1205931 (119904) = 0
1205932 (119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906119904 119907119904)
minus 119865 (119883120574119905119906119907
119904 119884
2119906119907(119904) 119885
2119906119907(119904) 119906119904 119907119904)
(62)
Denote 1205880(119903) = (1 + |119909|2)119903 due to 119887 120590 119891 are Lipishtiz and
that they are of linear growth Γ isin C12
119887 |1205932(119904)| le 1205880(|119883
120574119905119906119907
119904minus
120574119905(119905)|) we have
119864[int
119905+120575
119905
(100381610038161003816100381610038161198841119906119907
(119904) minus1198842119906119907
(119904)10038161003816100381610038161003816
2
+100381610038161003816100381610038161198851119906119907
(119904) minus1198852119906119907
(119904)10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864[int
119905+120575
119905
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) 119889119904 | F119905]
le 119862120575119864[ sup119904isin[119905119905+120575]
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) | F119905]
le 1198621205752
(63)
Mathematical Problems in Engineering 9
So100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816
=10038161003816100381610038161003816119864 [(119884
1119906119907(119904) minus 119884
2119906119907(119904)) | F119905]
10038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
119864 [int
119905+120575
119905
(119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904))
minus119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]
100381610038161003816100381610038161003816100381610038161003816
le 119862119864[int
119905+120575
119905
[1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
+100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816] 119889119904 | F119905]
le 119862119864[int
119905+120575
119905
1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) 119889119904 | F119905]
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
le 11986212057532
(64)
Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation
minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]
1198840 (119905 + 120575) = 0
(65)
where 1198650(120574119905 119910 119911) = sup119906isin119880
inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as
1198840 (119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) (66)
Proof First we define a function as follows
1198651 (120574119905 119910 119911 119906) = inf119907isin119881
119865 (120574119905 119910 119911 119906 119907)
(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880
(67)
Consider the following equation
minus1198891198843119906
(119904) = 1198651 (120574119905 1198843119906
(119904) 1198853119906
(119904) 119906 (119904)) 119889119904
minus1198853119906
(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]
1198843119906
(119905 + 120575) = 0
(68)
according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906
1198853119906
) solving (68) Also
1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as for every 119906 isin U119905119905+120575
(69)
In fact according to the definition of 1198651 and Lemma 7 wehave
1198843119906
(119905) le 1198842119906119907
(119905) 119875-as for any 119907 isin V119905119905+120575
for every 119906 isin U119905119905+120575
(70)
On the other side we have the existence of a measurablefunction 119907
4 R timesR119889
times 119880 rarr 119881 such that
1198651 (120574119905 119910 119911 119906)
= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906
(71)
Set 1199074(119904) = 1199074(119884
3119906(119904) 119885
3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know
1199074isin V119905119905+120575 Therefore (1198843119906
1198853119906
) = (11988421199061199074
11988521199061199074
) which isdue to the uniqueness of the solution of (68) In particular1198843119906
(119905) = 11988421199061199074
(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-119886119904 for every 119906 isin U119905119905+120575
Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880
1198651(120574119905 119910 119911 119906) wealso derive
1198840 (119905) = esssup119906isinU119905119905+120575
1198843119906
(119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as
(72)
Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575
32 119875-as
(73)
where the constant 119862 is independent of the control processes119906 119907
Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as
100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
2
le 119862120575
119864 [int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904] le 119862120575
(74)
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Definition 1 A functional 119906 Λ 997891rarr R is Λ-continuous at120574119905 isin Λ if for any 120576 gt 0 there exists 120575 gt 0 such that for each120574119905isin Λ with 119889infin(120574119905 120574119905) lt 120575 one has |119906(120574119905) minus 119906(120574
119905)| lt 120576
119906 is said to be Λ-continuous if it is Λ-continuous at each120574119905 isin Λ
Definition 2 Let 119907 Λ 997891rarr R and 120574119905 isin Λ be given If thereexists 119901 isin R119889 such that
119907 (120574119909
119905) = 119907 (120574119905) + ⟨119901 119909⟩ + 119900 (|119909|) as 119909 997888rarr 0 119909 isin R
119889
(7)
Then we say that 119907 is (vertically) differentiable at 119905 and denotethe gradient of 119863119909119907(120574119905) = 119901 sdot 119907 is said to be verticallydifferentiable in Λ if 119863119909119907(120574119905) exists for each 120574119905 isin Λ TheHessian119863119909119909119907(120574119905) can be defined similarly It is an 119878(119889)-valuedfunction defined on Λ where 119878(119889) is the space of all 119889 times 119889
symmetric matrices
For each 120574119905 isin Λ we denote 120574119905119904(119903) = 120574119905(119903)1[0119905)(119903) +
120574119905(119905)1[119905119904](119903) 119903 isin [0 119904] It is clear that 120574119905119904 isin Λ 119904
Definition 3 For a given 120574119905 isin Λ if one has
119907 (120574119905119904) = 119907 (120574119905) + 119886 (119904 minus 119905) + 119900 (|119904 minus 119905|) as 119904 997888rarr 119905 119904 ge 119905
(8)
then we say that 119907(120574119905) is (horizontally) differentiable in 119905
at 120574119905 and denote 119863119905119907(120574119905) = 119886 119907 is said to be horizontallydifferentiable in Λ if119863119905119907(120574119905) exists for each 120574119905 isin Λ
Definition 4 Define C119895119896(Λ) as the set of function 119907 =
(119907(120574119905))120574119905isinΛ defined on Λ which are 119895 times horizontally and
119896 times vertically differentiable in Λ such that all thesederivatives are Λ-continuous
The following is about the functional Itorsquos formula whichwas firstly obtained by Dupire [15] and then developed byCont and Fournie [18] for a more general formulation
Theorem 5 (functional Itorsquos formula) Let (ΩF (F119905)119905isin[0119879]
119875) be a probability space if 119883 is a continuous semimartingaleand 119907 is in C12
(Λ) then for any 119905 isin [0 119879)
119907 (119883119905) minus 119907 (1198830) = int
119905
0
119863119904119907 (119883119904) 119889119904 + int
119905
0
119863119909119907 (119883119904) 119889119883 (119904)
+1
2int
119905
0
119863119909119909119907 (119883119904) 119889 ⟨119883⟩ (119904) 119875-as
(9)
22 BSDEs In this section we recollect some importantresults which will be used in our SDG problems
Let (ΩF 119875) be the Wiener space where Ω is the set ofcontinuous functions from [0 119879] to R119889 starting from 0 (Ω =
1198620([0 119879]R119889)) 119879 is an arbitrarily fixed real time horizonF
is the completed Borel 120590-algebra overΩ and 119875 is the Wienermeasure Let 119861 be the canonical process 119861(120596 119904) = 120596119904 119904 isin
[0 119879] 120596 isin Ω We denote by F = F119904 0 le 119904 le 119879 the natural
filtration generated by 119861(119905)119905ge0 and augmented by all 119875-nullsets that is F119904 = 120590119861(119903) 119903 le 119904 or N119875 119904 isin [0 119879] F119904
119905=
120590(119861(119903)minus119861(119905) 119905 le 119903 le 119904)orN119875 whereN119875 is the set of all119875-nullsubsets First we present two spaces of processes as follows
S2(0 119879R
119899) = (120595(119905))
0le119905le119879R
119899-valued
F-adapted continuous process
119864 [ sup0le119905le119879
1003816100381610038161003816120595 (119905)10038161003816100381610038162] lt +infin
H2(0 119879R
119899) = (120595(119905))
0le119905le119879
R119899-valued
F-progressively measurable process
100381710038171003817100381712059510038171003817100381710038172= 119864[int
119879
0
1003816100381610038161003816120595 (119905)10038161003816100381610038162119889119905] lt +infin
(10)
Consider 119892 Ω times [0 119879] times R times R119889rarr R for every (119910 119911)
in R times R119889 (119892(119905 119910 119911))119905isin[0119879] is progressively measurable andsatisfies the following conditions
(A1) there exists a constant 119862 ge 0 such that 119875-as for all119905 isin [0 119879] 119910
1 1199102 isin R 1199111 1199112 isin R119889
1003816100381610038161003816119892 (119905 1199101 1199111) minus 119892 (119905 1199102 1199112)1003816100381610038161003816 le 119862 (
10038161003816100381610038161199101 minus 11991021003816100381610038161003816 +
10038161003816100381610038161199111 minus 11991121003816100381610038161003816) (11)
(A2) 119892(sdot 0 0) isin H2(0 119879R)
In the following we suppose the driver 119892 of a BSDEsatisfies (A1) and (A2)
Lemma 6 Under the assumptions (A1) and (A2) for anyrandom variable 120585 isin 119871
2(ΩF119879 119875R) the BSDE
119910 (119905) = 120585 + int
119879
119905
119892 (119904 119910 (119904) 119911 (119904)) 119889119904 minus int
119879
119905
119911 (119904) 119889119861 (119904)
0 le 119905 le 119879
(12)
has a unique adapted solution
(119910(119905) 119911(119905))119905isin[0119879]
isin S2(0 119879R) timesH
2(0 119879R
119889) (13)
The readers may refer to Pardoux and Peng [1] forthe above well-known existence and uniqueness results onBSDEs
Lemma 7 (comparison theorem) Given two coefficients 11989211198922 satisfying (A1) (A2) and two terminal values 1205851 1205852 isin
1198712(ΩF119879 119875R) by (119910
1 119911
1) and (119910
2 119911
2) one denotes the
solution of a BSDE with the data (1205851 1198921) and (1205852 1198922)respectively Then one has the following
(i) If 1205851 ge 1205852 and 1198921 ge 1198922 P-as then 1199101(119905) ge 119910
2(119905)
P-as for all 119905 isin [0 119879]
4 Mathematical Problems in Engineering
(ii) (Strict monotonicity) If in addition to (i) one alsoassumes that 119875(1205851 gt 1205852) gt 0 then 119875(119910
1(119905) gt 119910
2(119905)) gt
0 0 le 119905 le 119879 and in particular 1199101(0) gt 119910
2(0)
With the notations in Lemma 7 we assume that for some119892 Ω times [0 119879] times R times R119889
rarr R satisfying (A1) and (A2) thedrivers 119892119894 have the following form
119892119894 (119904 119910119894(119904) 119911
119894(119904)) = 119892 (119904 119910
119894(119904) 119911
119894(119904)) + 120593119894 (119904)
119889119904119889119875-ae 119894 = 1 2
(14)
where 120593119894 isin H2(0 119879R) Then the following results hold true
for all terminal values 1205851 1205852 isin 1198712(ΩF119879 119875R)
Lemma 8 The difference of the solutions (1199101 119911
1) and (119910
2 119911
2)
of BSDE (12) with the data (1205851 1198921) and (1205852 1198922) respectivelysatisfies the following estimate
100381610038161003816100381610038161199101(119905)minus119910
2(119905)
10038161003816100381610038161003816
2
+1
2119864 [int
119879
119905
119890120573(119904minus119905)
(100381610038161003816100381610038161199101(119904)minus119910
2(119904)
10038161003816100381610038161003816
2
+100381610038161003816100381610038161199111(119904) minus 119911
2(119904)
10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864 [119890120573(119879minus119905)10038161003816100381610038161205851 minus 1205852
10038161003816100381610038162| F119905]
+ 119864 [int
119879
119905
119890120573(119904minus119905)10038161003816100381610038161205931 (119904) minus 1205932 (119904)
10038161003816100381610038162119889119904 | F119905]
P-as forall0 le 119905 le 119879 where 120573 = 16 (1 + 1198622)
(15)
Proof The reader may refer to Proposition 21 in El Karoui etal [5] or Theorem 23 in Peng [9] for the details
3 A DPP for Stochastic Differential Games ofFunctional FBSDEs
In this section we consider the SDGs of functional FBSDEsFirst we introduce the background of SDGs Suppose
that the control state spaces 119880119881 are compact metric spacesU (resp V) is the control set of all 119880 (resp 119881)-valuedF-progressively measurable processes for the first (respsecond) player If 119906 isin U (resp 119907 isin V) we call 119906 (resp 119907)an admissible control
Let us give the following mappings
119887 Λ times 119880 times 119881 997888rarr R119899
120590 Λ times 119880 times 119881 997888rarr R119899times119889
119891 Λ timesR timesR119889times 119880 times 119881 997888rarr R
(16)
For given admissible controls 119906(sdot) isin U 119907(sdot) isin V and 119905 isin
[0 119879] 120574119905 isin Λ we consider the following functional forward-backward stochastic system
119889119883120574119905119906119907
(119904) = 119887 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119904
+ 120590 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]
119889119884120574119905119906119907
(119904) = minus119891 (119883120574119905119906119907
119904 119884
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
119883120574119905119906119907
119905= 120574119905
119884120574119905119906119907
(119879) = Φ (119883120574119905119906119907
119879)
(17)
(H) (i) For all 119905 isin [0 119879] 119906 isin 119880 119907 isin 119881 119909119905 isin Λ 119910 isin
R 119911 isin R119889 119887(119909119905 119906 119907) 120590(119909119905 119906 119907) and 119891(119909119905 119910 119911 119906 119907)
areF119905-measurable(ii) There exists a constant 119862 gt 0 such that for all 119905 isin
[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091
119905 119909
2
119905isin Λ
10038161003816100381610038161003816119887 (119909
1
119905 119906 119907) minus 119887 (119909
2
119905 119906 119907)
10038161003816100381610038161003816+10038161003816100381610038161003816120590 (119909
1
119905 119906 119907) minus 120590 (119909
2
119905 119906 119907)
10038161003816100381610038161003816
le 119862100381710038171003817100381710038171199091
119905minus 119909
2
119905
10038171003817100381710038171003817
(18)1003816100381610038161003816119887 (119909119905 119906 119907)
1003816100381610038161003816 +1003816100381610038161003816120590 (119909119905 119906 119907)
1003816100381610038161003816 le 119862 (1 +1003817100381710038171003817119909119905
1003817100381710038171003817) for any 119909119905 isin Λ
(19)
(iii) There exists a constant 119862 gt 0 such that for all 119905 isin
[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091
119905 119909
2
119905isin Λ 119910
1 119910
2isin
R 1199111 119911
2isin R119889
10038161003816100381610038161003816119891 (119909
1
119905 119910
1 119911
1 119906 119907) minus 119891 (119909
2
119905 119910
2 119911
2 119906 119907)
10038161003816100381610038161003816
le 119862 (100381710038171003817100381710038171199091
119905minus 119909
2
119905
10038171003817100381710038171003817+100381610038161003816100381610038161199101minus 119910
210038161003816100381610038161003816+100381610038161003816100381610038161199111minus 119911
210038161003816100381610038161003816)
10038161003816100381610038161003816Φ (119909
1
119879) minus Φ (119909
2
119879)10038161003816100381610038161003816le 119862
100381710038171003817100381710038171199091
119879minus 119909
2
119879
10038171003817100381710038171003817
1003816100381610038161003816119891 (119909119905 0 0 119906 119907)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171199091199051003817100381710038171003817)
1003816100381610038161003816Φ (119909119879)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171199091198791003817100381710038171003817) for any 119909119905 isin Λ
(20)
Theorem 9 Under the assumption (H) there exists a uniquesolution (119883 119884 119885) isin S2
(0 119879R119899) timesS2
(0 119879R) timesH2(0 119879R119889
)
solving (17)
We recall the subspaces of admissible controls and thedefinitions of admissible strategies as follows which aresimilar to [10]
Definition 10 An admissible control process 119906 = (119906119903)119903isin[119905119904]
(resp 119907 = (119907119903)119903isin[119905119904]) for Player I (resp II) on [119905 119904] is anF119903-progressively measurable 119880 (resp 119881)-valued process Theset of all admissible controls for Player I (resp II) on [119905 119904]
is denoted byU119905119904 (respV119905119904) If 119875119906 equiv 119906 119886119890 119894119899 [119905 119904] = 1one will identify both processes 119906 and 119906 inU119905119904 Similarly oneinterprets 119907 equiv 119907 on [119905 119904] inV119905119904
Mathematical Problems in Engineering 5
Definition 11 A nonanticipative strategy for Player I on[119905 119904] (119905 lt 119904 le 119879) is a mapping 120572 V119905119904 rarr U119905119904 suchthat for any F-stopping time 119878 Ω rarr [119905 119904] and any1199071 1199072 isin V119905119904 with 1199071 equiv 1199072 on [[119905 119878]] it holds that 120572(1199071) equiv
120572(1199072) on [[119905 119878]] Nonanticipative strategies for Player II on[119905 119904] 120573 U119905119904 rarr V119905119904 are defined similarly The set ofall nonanticipative strategies 120572 V119905119904 rarr U119905119904 for PlayerI on [119905 119904] is denoted by A119905119904 The set of all nonanticipativestrategies 120573 U119905119904 rarr V119905119904 for Player II on [119905 119904] is denoted byB119905119904 (Recall that [[119905 119878]] = (119903 120596) isin [0 119879] times Ω 119905 le 119903 le 119878(120596))
For given processes 119906(sdot) isin U119905119879 119907(sdot) isin V119905119879 initial data119905 isin [0 119879] 120574119905 isin Λ the cost functional is defined as follows
119869 (120574119905 119906 119907) = 119884120574119905119906119907
(119905) 120574119905 isin Λ (21)
where the process 119884120574119905119906119907 is defined by functional FBSDE (17)
For 120574119905 isin Λ the lower and the upper value functions of ourSDGs are defined as
119882(120574119905) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (22)
119880 (120574119905) = esssup120572isinA119905119879
essinf119907isinV119905119879
119869 (120574119905 120572 (119907) 119907) (23)
As we know the essential infimum and essential supre-mum on a family of random variables are still random vari-ables But by applying the method introduced by Buckdahnand Li [10] we get119882(120574119905) and 119880(120574119905) are deterministic
Proposition 12 For any 119905 isin [0 119879] 120574119905 isin Λ 119882(120574119905) is a deter-ministic function in the sense that119882(120574119905) = 119864[119882(120574119905)] 119875-as
Proof Let 119867 denote the Cameron-Martin space of all abso-lutely continuous elements ℎ isin Ω whose derivative ℎ belongsto 119871
2([0 119879]R119889
)For any ℎ isin 119867 we define the mapping 120591ℎ120596 = 120596 + ℎ 120596 isin
Ω It is easy to check that 120591ℎ Ω rarr Ω is a bijectionand its law is given by 119875 ∘ [120591ℎ]
minus1= expint119879
0ℎ(119904)119889119861(119904) minus
(12) int119879
0|ℎ(119904)|
2119889119904119875 For any fixed 119905 isin [0 119879] set 119867119905 = ℎ isin
119867 | ℎ(sdot) = ℎ(sdot and 119905) The proof can be separated into thefollowing four steps
(1) For all 119906 isin U119905119879 119907 isin V119905119879 ℎ isin 119867119905 119869(120574119905 119906 119907)(120591ℎ) =
119869(120574119905 119906(120591ℎ) 119907(120591ℎ)) 119875-asFirst we make the transformation for the functional SDE
as follows
119883120574119905119906119907
(119904)∘(120591ℎ)= (120574119905 (119905)+int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
+int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)
= 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
(24)
then from the uniqueness of the solution of the functionalSDE we get
119883120574119905119906119907
(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as(25)
Similarly using the transformation to the BSDE in (17) andcomparing the obtained equation with the BSDE obtainedfrom (17) by replacing the transformed control process119906(120591ℎ) 119907(120591ℎ) for 119906 119907 due to the uniqueness of the solution offunctional BSDE we obtain
119884120574119905119906119907
(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as
119885120574119905119906119907
(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)
119889119904119889119875-ae on [0 119879] times Ω
(26)
Hence
119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)
(2) For 120573 isin B119905119879 ℎ isin 119867119905 let 120573ℎ(119906) = 120573(119906(120591minusℎ))(120591ℎ) 119906 isin
U119905119879 Then 120573ℎisin B119905119879
Obviously 120573ℎ U119905119879 rarr V119905119879 And it is nonanticipating
In fact given an F-stopping time 119878 Ω rarr [119905 119879] and 1199061 1199062 isin
U119905119879 with 1199061 equiv 1199062 on [[119905 119878]] Accordingly 1199061(120591minusℎ) equiv 1199062(120591minusℎ)
on [[119905 119878(120591minusℎ)]] Thus
120573ℎ(1199061) = 120573 (1199061 (120591minusℎ)) (120591ℎ) = 120573 (1199062 (120591minusℎ)) (120591ℎ)
= 120573ℎ(1199062) on [[119905 119878]]
(28)
(3) For any ℎ isin 119867119905 and 120573 isin B119905119879 we have
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)
In fact for convenience setting 119868(120574119905 120573) = esssup119906isinU119905119879
119869(120574119905 119906
120573(119906)) 120573 isin B119905119879 we know 119868(120574119905 120573) ge 119869(120574119905 119906 120573(119906)) Then
6 Mathematical Problems in Engineering
119868(120574119905 120573)(120591ℎ) ge 119869(120574119905 119906 120573(119906))(120591ℎ) 119875-as for all 119906 isin U119905119879Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)
From the definition of essential supremum for anyrandom variable 120585 which satisfies 120585 ge 119869(120574119905 119906 120573(119906))(120591ℎ) wehave 120585(120591minusℎ) ge 119869(120574119905 119906 120573(119906)) 119875-as for all 119906 isin U119905119879 So120585(120591minusℎ) ge 119868(120574119905 120573) 119875-as that is 120585 ge 119868(120574119905 120573)(120591ℎ) 119875-as Thus
119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
119875-as for any 119906 isin U119905119879
(31)
Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)
From above we get
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)
(4) Under the Girsanov transformation 120591ℎ 119882(120574119905) isinvariant that is
119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)
In fact we can prove essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) =
essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) 119875-as for all ℎ isin 119867119905 which issimilar to the above step From the above three steps for allℎ isin 119867119905 we get
119882(120574119905) (120591ℎ) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906))
= 119882 (120574119905) 119875-as(35)
Note that 119906(120591ℎ) | 119906(sdot) isin U119905119879 = U119905119879 and 120573ℎ
| 120573 isin
B119905119879 = B119905119879 have been used in the above equalities So forany ℎ isin 119867119905119882(120574119905)(120591ℎ) = 119882(120574119905) 119875-as Thanks to 119882(120574119905) isF119905-measurable this relation holds true for all ℎ isin 119867
To finish the proof we also need the auxiliary lemma asfollows
Lemma 13 Let 120577 be a random variable defined over theclassical Wiener space (ΩF119879 119875) such that 120577(120591ℎ) = 120577 119875-asfor any ℎ isin 119867 Then 120577 = 119864120577 119875-as
Proof From Lemma 13 in Buckdahn and Li [10] we know forany 119860 isin B(R) 120593 isin 119871
2([0 119879]R119889
)
119864[1120577isin119860 expint
119879
0
120593 (119904) 119889119861 (119904)]
= 119864 [1120577isin119860] 119864 [expint
119879
0
120593 (119904) 119889119861 (119904)]
(36)
For any 120593 = sum119873
119894=11205931198941(119905
119894minus1119905119894] where 120593119894 isin R119889 for 0 le 119894 le 119873 and
119905119894119873
119894=0is a finite partition of [0 119879] from (36)
119864[1120577isin119860 exp(
119873
sum
119894=1
120593119894119861 (119905119894) minus 119861 (119905119894minus1))]
= 119864 [1120577isin119860] 119864[exp119873
sum
119894=1
120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]
(37)
Therefore for any nonnegative integer 119896119894
119864[1120577isin119860
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
= 119864 [1120577isin119860] 119864[
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
(38)
So for any polynomial function 119876 we have
119864 [1120577isin119860119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
= 119864 [1120577isin119860] 119864 [119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
(39)
Furthermore for any 119876 isin 119862119887(R119899) we still have (39)
Combining the arbitrariness of 119860 isin B(R) we obtain 120577 isindependent of (119861(1199051) minus 119861(1199050) 119861(119905119899) minus 119861(119905119899minus1)) for allpartition of [0 119879] Therefore 120577 is independent of F119879 whichimplies 120577 is independent of itself that is 119864120577 = 120577 119875-as
In Ji and Yang [14] they proved the following estimates
Mathematical Problems in Engineering 7
Lemma 14 Under the assumption (H) there exists someconstant 119862 gt 0 such that for any 119905 isin [0 119879] 120574119905 120574119905 isin Λ 119906(sdot) isin
U 119907(sdot) isin V
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905]le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119883
120574119905119906119907
(119904) minus 119883120574119905119906119907
(119904)10038161003816100381610038161003816
2
| F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
119864 [ sup119904isin[119905119879]
1003816100381610038161003816119884120574119905119906119907
(119904)10038161003816100381610038162+int
119879
119905
1003816100381610038161003816119885120574119905119906119907
(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119884120574119905119906119907
(119904) minus 119884120574119905119906119907
(119904)10038161003816100381610038161003816
2
+int
119879
119905
10038161003816100381610038161003816119885120574119905119906119907
(119904) minus 119885120574119905119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
(40)From the definition of119882(120574119905) and Lemma 14 we have the
following property
Lemma 15 There exists some constant 119862 gt 0 such that for all0 le 119905 le 119879 120574119905 120574119905 isin Λ
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817
(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171205741199051003817100381710038171003817)
(41)
Nowwe adopt Pengrsquos notion of stochastic backward semi-group (which was first introduced by Peng [9] to prove theDPP for stochastic control problems) to discuss a generalizedDPP for our SDG (17) (22) First we define the family ofbackward semigroups associated with FBSDE (17)
For given 119905 isin [0 119879] 120574119905 isin Λ a number 120575 isin (0 119879 minus 119905]admissible control processes 119906(sdot) isin U119905119905+120575 119907(sdot) isin V119905119905+120575 weset
119866120574119905119906119907
119904119905+120575[120578] =
120574119905119906119907
(119904) 119904 isin [119905 119905 + 120575] (42)
where 120578 isin 1198712(ΩF119905+120575
119905 119875R) (
120574119905119906119907
(119904) 119885120574119905119906119907
(119904))119905le119904le119905+120575
solves the following functional FBSDE on [119905 119905 + 120575]
119889120574119905119906119907
(119904)
= minus119891 (119904 119883120574119905119906119907
119904
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
120574119905119906119907
(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]
(43)Also we have
119866120574119905119906119907
119905119879[Φ (119883
120574119905119906119907
119879)] = 119866
120574119905119906119907
119905119905+120575[119884
120574119905119906119907
(119905 + 120575)] (44)Theorem 16 Suppose (H) holds true the lower value function119882(120574119905) satisfies the followingDPP for any 119905 isin [0 119879] 120574119905 isin Λ 120575 gt
0
119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (45)
The proof is given in the appendix
4 Viscosity Solutions of Path-DependentHJBI Equation
Now we study the following path-dependent PDEs
119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0
119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ
(46)
119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0
119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ
(47)
where
119867minus(120574119905119882119863119909119882119863119909119909119882)
= sup119906isin119880
inf119907isin119881
119867(120574119905119882119863119909119882119863119909119909119882119906 119907)
119867+(120574119905 119880119863119909119880119863119909119909119880)
= inf119907isin119881
sup119906isin119880
119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)
119867 (120574119905 119910 119901 119883 119906 119907)
=1
2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)
+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)
(48)
where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of
119889 times 119889 symmetric matrices)We will show that the value function119882(120574119905) (resp 119880(120574119905))
defined in (22) (resp (23)) is a viscosity solution of thecorresponding equation (46) (resp (47)) First we give thedefinition of viscosity solution for this kind of PDEs Formore information on viscosity solution the reader is referredto Crandall et al [22]
Definition 17 A real-valued Λ-continuous function 119882 isin
C(Λ) is called
(i) a viscosity subsolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ ge 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)
(ii) a viscosity supersolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ le 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)
(iii) a viscosity solution of (46) if it is both a viscosity sub-and supersolution of (46)
Theorem 18 Assume (H) holds the lower value function119882 isa viscosity solution of path-dependent HJBI Equation (46) onΛ the upper value function 119880 is a viscosity solution of (47)
8 Mathematical Problems in Engineering
First we prove some helpful lemmas For some fixed Γ isin
C12
119897119887(Λ) denote
119865 (120574119904 119910 119911 119906 119907)
= 119863119904Γ (120574119904) +1
2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))
+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)
+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)
(51)
where (120574119904 119910 119911 119906 119907) isin Λ timesR timesR119889times 119880 times 119881
Consider the following BSDE
minus 1198891198841119906119907
(119904)
= 119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198851119906119907
(119904) 119889119861 (119904)
1198841119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]
(52)
Lemma 19 For every 119904 isin [119905 119905 + 120575] one has the following
1198841119906119907
(119904) = 119866120574119905119906119907
119904119905+120575[Γ (119883
120574119905119906119907
119905+120575)] minus Γ (119883
120574119905119906119907
119904) 119875-as (53)
Proof Note119866120574119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] is defined as119866120574
119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] =
119884119906119907
(119904) through the following BSDE
minus 119889119884119906119907
(119904) = 119891 (119883120574119905119906119907
119904 119884
119906119907(119904) 119885
119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 119885119906119907
(119904) 119889119861 (119904)
119884119906119907
(119905 + 120575) = Γ (119883120574119905119906119907
119905+120575) 119904 isin [119905 119905 + 120575]
(54)
Using Itorsquos formula to Γ(119883120574119905119906119907
119904) we have
119889 (119884119906119907
(119904) minus Γ (119883120574119905119906119907
119904)) = 119889119884
1119906119907(119904) (55)
Combined with 119884119906119907
(119905 + 120575) minus Γ(119883120574119905119906119907
119905+120575) = 0 = 119884
1119906119907(119905 + 120575) we
get the desired result
Now consider the following BSDE
minus 1198891198842119906119907
(119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198852119906119907
(119904) 119889119861 (119904)
(56)
1198842119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)
Then we have the following lemma
Lemma 20 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816le 119862120575
32 119875-as (58)
where 119862 is independent of the control processes 119906 119907
Proof From Lemma 14 we know there exists some constant119862 gt 0 such that
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905] le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172) (59)
combined with
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]
le 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
10038161003816100381610038161003816100381610038161003816
2
| F119905]
+ 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)
10038161003816100381610038161003816100381610038161003816
2
| F119905]
(60)
we have
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)
From (52) and (54) using Lemma 8 set
1205851 = 1205852 = 0
119892 (119904 119910 119911) = 119865 (119883120574119905119906119907
119904 119910 119911 119906119904 119907119904)
1205931 (119904) = 0
1205932 (119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906119904 119907119904)
minus 119865 (119883120574119905119906119907
119904 119884
2119906119907(119904) 119885
2119906119907(119904) 119906119904 119907119904)
(62)
Denote 1205880(119903) = (1 + |119909|2)119903 due to 119887 120590 119891 are Lipishtiz and
that they are of linear growth Γ isin C12
119887 |1205932(119904)| le 1205880(|119883
120574119905119906119907
119904minus
120574119905(119905)|) we have
119864[int
119905+120575
119905
(100381610038161003816100381610038161198841119906119907
(119904) minus1198842119906119907
(119904)10038161003816100381610038161003816
2
+100381610038161003816100381610038161198851119906119907
(119904) minus1198852119906119907
(119904)10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864[int
119905+120575
119905
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) 119889119904 | F119905]
le 119862120575119864[ sup119904isin[119905119905+120575]
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) | F119905]
le 1198621205752
(63)
Mathematical Problems in Engineering 9
So100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816
=10038161003816100381610038161003816119864 [(119884
1119906119907(119904) minus 119884
2119906119907(119904)) | F119905]
10038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
119864 [int
119905+120575
119905
(119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904))
minus119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]
100381610038161003816100381610038161003816100381610038161003816
le 119862119864[int
119905+120575
119905
[1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
+100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816] 119889119904 | F119905]
le 119862119864[int
119905+120575
119905
1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) 119889119904 | F119905]
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
le 11986212057532
(64)
Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation
minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]
1198840 (119905 + 120575) = 0
(65)
where 1198650(120574119905 119910 119911) = sup119906isin119880
inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as
1198840 (119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) (66)
Proof First we define a function as follows
1198651 (120574119905 119910 119911 119906) = inf119907isin119881
119865 (120574119905 119910 119911 119906 119907)
(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880
(67)
Consider the following equation
minus1198891198843119906
(119904) = 1198651 (120574119905 1198843119906
(119904) 1198853119906
(119904) 119906 (119904)) 119889119904
minus1198853119906
(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]
1198843119906
(119905 + 120575) = 0
(68)
according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906
1198853119906
) solving (68) Also
1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as for every 119906 isin U119905119905+120575
(69)
In fact according to the definition of 1198651 and Lemma 7 wehave
1198843119906
(119905) le 1198842119906119907
(119905) 119875-as for any 119907 isin V119905119905+120575
for every 119906 isin U119905119905+120575
(70)
On the other side we have the existence of a measurablefunction 119907
4 R timesR119889
times 119880 rarr 119881 such that
1198651 (120574119905 119910 119911 119906)
= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906
(71)
Set 1199074(119904) = 1199074(119884
3119906(119904) 119885
3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know
1199074isin V119905119905+120575 Therefore (1198843119906
1198853119906
) = (11988421199061199074
11988521199061199074
) which isdue to the uniqueness of the solution of (68) In particular1198843119906
(119905) = 11988421199061199074
(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-119886119904 for every 119906 isin U119905119905+120575
Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880
1198651(120574119905 119910 119911 119906) wealso derive
1198840 (119905) = esssup119906isinU119905119905+120575
1198843119906
(119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as
(72)
Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575
32 119875-as
(73)
where the constant 119862 is independent of the control processes119906 119907
Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as
100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
2
le 119862120575
119864 [int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904] le 119862120575
(74)
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
(ii) (Strict monotonicity) If in addition to (i) one alsoassumes that 119875(1205851 gt 1205852) gt 0 then 119875(119910
1(119905) gt 119910
2(119905)) gt
0 0 le 119905 le 119879 and in particular 1199101(0) gt 119910
2(0)
With the notations in Lemma 7 we assume that for some119892 Ω times [0 119879] times R times R119889
rarr R satisfying (A1) and (A2) thedrivers 119892119894 have the following form
119892119894 (119904 119910119894(119904) 119911
119894(119904)) = 119892 (119904 119910
119894(119904) 119911
119894(119904)) + 120593119894 (119904)
119889119904119889119875-ae 119894 = 1 2
(14)
where 120593119894 isin H2(0 119879R) Then the following results hold true
for all terminal values 1205851 1205852 isin 1198712(ΩF119879 119875R)
Lemma 8 The difference of the solutions (1199101 119911
1) and (119910
2 119911
2)
of BSDE (12) with the data (1205851 1198921) and (1205852 1198922) respectivelysatisfies the following estimate
100381610038161003816100381610038161199101(119905)minus119910
2(119905)
10038161003816100381610038161003816
2
+1
2119864 [int
119879
119905
119890120573(119904minus119905)
(100381610038161003816100381610038161199101(119904)minus119910
2(119904)
10038161003816100381610038161003816
2
+100381610038161003816100381610038161199111(119904) minus 119911
2(119904)
10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864 [119890120573(119879minus119905)10038161003816100381610038161205851 minus 1205852
10038161003816100381610038162| F119905]
+ 119864 [int
119879
119905
119890120573(119904minus119905)10038161003816100381610038161205931 (119904) minus 1205932 (119904)
10038161003816100381610038162119889119904 | F119905]
P-as forall0 le 119905 le 119879 where 120573 = 16 (1 + 1198622)
(15)
Proof The reader may refer to Proposition 21 in El Karoui etal [5] or Theorem 23 in Peng [9] for the details
3 A DPP for Stochastic Differential Games ofFunctional FBSDEs
In this section we consider the SDGs of functional FBSDEsFirst we introduce the background of SDGs Suppose
that the control state spaces 119880119881 are compact metric spacesU (resp V) is the control set of all 119880 (resp 119881)-valuedF-progressively measurable processes for the first (respsecond) player If 119906 isin U (resp 119907 isin V) we call 119906 (resp 119907)an admissible control
Let us give the following mappings
119887 Λ times 119880 times 119881 997888rarr R119899
120590 Λ times 119880 times 119881 997888rarr R119899times119889
119891 Λ timesR timesR119889times 119880 times 119881 997888rarr R
(16)
For given admissible controls 119906(sdot) isin U 119907(sdot) isin V and 119905 isin
[0 119879] 120574119905 isin Λ we consider the following functional forward-backward stochastic system
119889119883120574119905119906119907
(119904) = 119887 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119904
+ 120590 (119883120574119905119906119907
119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]
119889119884120574119905119906119907
(119904) = minus119891 (119883120574119905119906119907
119904 119884
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
119883120574119905119906119907
119905= 120574119905
119884120574119905119906119907
(119879) = Φ (119883120574119905119906119907
119879)
(17)
(H) (i) For all 119905 isin [0 119879] 119906 isin 119880 119907 isin 119881 119909119905 isin Λ 119910 isin
R 119911 isin R119889 119887(119909119905 119906 119907) 120590(119909119905 119906 119907) and 119891(119909119905 119910 119911 119906 119907)
areF119905-measurable(ii) There exists a constant 119862 gt 0 such that for all 119905 isin
[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091
119905 119909
2
119905isin Λ
10038161003816100381610038161003816119887 (119909
1
119905 119906 119907) minus 119887 (119909
2
119905 119906 119907)
10038161003816100381610038161003816+10038161003816100381610038161003816120590 (119909
1
119905 119906 119907) minus 120590 (119909
2
119905 119906 119907)
10038161003816100381610038161003816
le 119862100381710038171003817100381710038171199091
119905minus 119909
2
119905
10038171003817100381710038171003817
(18)1003816100381610038161003816119887 (119909119905 119906 119907)
1003816100381610038161003816 +1003816100381610038161003816120590 (119909119905 119906 119907)
1003816100381610038161003816 le 119862 (1 +1003817100381710038171003817119909119905
1003817100381710038171003817) for any 119909119905 isin Λ
(19)
(iii) There exists a constant 119862 gt 0 such that for all 119905 isin
[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091
119905 119909
2
119905isin Λ 119910
1 119910
2isin
R 1199111 119911
2isin R119889
10038161003816100381610038161003816119891 (119909
1
119905 119910
1 119911
1 119906 119907) minus 119891 (119909
2
119905 119910
2 119911
2 119906 119907)
10038161003816100381610038161003816
le 119862 (100381710038171003817100381710038171199091
119905minus 119909
2
119905
10038171003817100381710038171003817+100381610038161003816100381610038161199101minus 119910
210038161003816100381610038161003816+100381610038161003816100381610038161199111minus 119911
210038161003816100381610038161003816)
10038161003816100381610038161003816Φ (119909
1
119879) minus Φ (119909
2
119879)10038161003816100381610038161003816le 119862
100381710038171003817100381710038171199091
119879minus 119909
2
119879
10038171003817100381710038171003817
1003816100381610038161003816119891 (119909119905 0 0 119906 119907)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171199091199051003817100381710038171003817)
1003816100381610038161003816Φ (119909119879)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171199091198791003817100381710038171003817) for any 119909119905 isin Λ
(20)
Theorem 9 Under the assumption (H) there exists a uniquesolution (119883 119884 119885) isin S2
(0 119879R119899) timesS2
(0 119879R) timesH2(0 119879R119889
)
solving (17)
We recall the subspaces of admissible controls and thedefinitions of admissible strategies as follows which aresimilar to [10]
Definition 10 An admissible control process 119906 = (119906119903)119903isin[119905119904]
(resp 119907 = (119907119903)119903isin[119905119904]) for Player I (resp II) on [119905 119904] is anF119903-progressively measurable 119880 (resp 119881)-valued process Theset of all admissible controls for Player I (resp II) on [119905 119904]
is denoted byU119905119904 (respV119905119904) If 119875119906 equiv 119906 119886119890 119894119899 [119905 119904] = 1one will identify both processes 119906 and 119906 inU119905119904 Similarly oneinterprets 119907 equiv 119907 on [119905 119904] inV119905119904
Mathematical Problems in Engineering 5
Definition 11 A nonanticipative strategy for Player I on[119905 119904] (119905 lt 119904 le 119879) is a mapping 120572 V119905119904 rarr U119905119904 suchthat for any F-stopping time 119878 Ω rarr [119905 119904] and any1199071 1199072 isin V119905119904 with 1199071 equiv 1199072 on [[119905 119878]] it holds that 120572(1199071) equiv
120572(1199072) on [[119905 119878]] Nonanticipative strategies for Player II on[119905 119904] 120573 U119905119904 rarr V119905119904 are defined similarly The set ofall nonanticipative strategies 120572 V119905119904 rarr U119905119904 for PlayerI on [119905 119904] is denoted by A119905119904 The set of all nonanticipativestrategies 120573 U119905119904 rarr V119905119904 for Player II on [119905 119904] is denoted byB119905119904 (Recall that [[119905 119878]] = (119903 120596) isin [0 119879] times Ω 119905 le 119903 le 119878(120596))
For given processes 119906(sdot) isin U119905119879 119907(sdot) isin V119905119879 initial data119905 isin [0 119879] 120574119905 isin Λ the cost functional is defined as follows
119869 (120574119905 119906 119907) = 119884120574119905119906119907
(119905) 120574119905 isin Λ (21)
where the process 119884120574119905119906119907 is defined by functional FBSDE (17)
For 120574119905 isin Λ the lower and the upper value functions of ourSDGs are defined as
119882(120574119905) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (22)
119880 (120574119905) = esssup120572isinA119905119879
essinf119907isinV119905119879
119869 (120574119905 120572 (119907) 119907) (23)
As we know the essential infimum and essential supre-mum on a family of random variables are still random vari-ables But by applying the method introduced by Buckdahnand Li [10] we get119882(120574119905) and 119880(120574119905) are deterministic
Proposition 12 For any 119905 isin [0 119879] 120574119905 isin Λ 119882(120574119905) is a deter-ministic function in the sense that119882(120574119905) = 119864[119882(120574119905)] 119875-as
Proof Let 119867 denote the Cameron-Martin space of all abso-lutely continuous elements ℎ isin Ω whose derivative ℎ belongsto 119871
2([0 119879]R119889
)For any ℎ isin 119867 we define the mapping 120591ℎ120596 = 120596 + ℎ 120596 isin
Ω It is easy to check that 120591ℎ Ω rarr Ω is a bijectionand its law is given by 119875 ∘ [120591ℎ]
minus1= expint119879
0ℎ(119904)119889119861(119904) minus
(12) int119879
0|ℎ(119904)|
2119889119904119875 For any fixed 119905 isin [0 119879] set 119867119905 = ℎ isin
119867 | ℎ(sdot) = ℎ(sdot and 119905) The proof can be separated into thefollowing four steps
(1) For all 119906 isin U119905119879 119907 isin V119905119879 ℎ isin 119867119905 119869(120574119905 119906 119907)(120591ℎ) =
119869(120574119905 119906(120591ℎ) 119907(120591ℎ)) 119875-asFirst we make the transformation for the functional SDE
as follows
119883120574119905119906119907
(119904)∘(120591ℎ)= (120574119905 (119905)+int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
+int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)
= 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
(24)
then from the uniqueness of the solution of the functionalSDE we get
119883120574119905119906119907
(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as(25)
Similarly using the transformation to the BSDE in (17) andcomparing the obtained equation with the BSDE obtainedfrom (17) by replacing the transformed control process119906(120591ℎ) 119907(120591ℎ) for 119906 119907 due to the uniqueness of the solution offunctional BSDE we obtain
119884120574119905119906119907
(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as
119885120574119905119906119907
(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)
119889119904119889119875-ae on [0 119879] times Ω
(26)
Hence
119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)
(2) For 120573 isin B119905119879 ℎ isin 119867119905 let 120573ℎ(119906) = 120573(119906(120591minusℎ))(120591ℎ) 119906 isin
U119905119879 Then 120573ℎisin B119905119879
Obviously 120573ℎ U119905119879 rarr V119905119879 And it is nonanticipating
In fact given an F-stopping time 119878 Ω rarr [119905 119879] and 1199061 1199062 isin
U119905119879 with 1199061 equiv 1199062 on [[119905 119878]] Accordingly 1199061(120591minusℎ) equiv 1199062(120591minusℎ)
on [[119905 119878(120591minusℎ)]] Thus
120573ℎ(1199061) = 120573 (1199061 (120591minusℎ)) (120591ℎ) = 120573 (1199062 (120591minusℎ)) (120591ℎ)
= 120573ℎ(1199062) on [[119905 119878]]
(28)
(3) For any ℎ isin 119867119905 and 120573 isin B119905119879 we have
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)
In fact for convenience setting 119868(120574119905 120573) = esssup119906isinU119905119879
119869(120574119905 119906
120573(119906)) 120573 isin B119905119879 we know 119868(120574119905 120573) ge 119869(120574119905 119906 120573(119906)) Then
6 Mathematical Problems in Engineering
119868(120574119905 120573)(120591ℎ) ge 119869(120574119905 119906 120573(119906))(120591ℎ) 119875-as for all 119906 isin U119905119879Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)
From the definition of essential supremum for anyrandom variable 120585 which satisfies 120585 ge 119869(120574119905 119906 120573(119906))(120591ℎ) wehave 120585(120591minusℎ) ge 119869(120574119905 119906 120573(119906)) 119875-as for all 119906 isin U119905119879 So120585(120591minusℎ) ge 119868(120574119905 120573) 119875-as that is 120585 ge 119868(120574119905 120573)(120591ℎ) 119875-as Thus
119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
119875-as for any 119906 isin U119905119879
(31)
Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)
From above we get
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)
(4) Under the Girsanov transformation 120591ℎ 119882(120574119905) isinvariant that is
119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)
In fact we can prove essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) =
essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) 119875-as for all ℎ isin 119867119905 which issimilar to the above step From the above three steps for allℎ isin 119867119905 we get
119882(120574119905) (120591ℎ) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906))
= 119882 (120574119905) 119875-as(35)
Note that 119906(120591ℎ) | 119906(sdot) isin U119905119879 = U119905119879 and 120573ℎ
| 120573 isin
B119905119879 = B119905119879 have been used in the above equalities So forany ℎ isin 119867119905119882(120574119905)(120591ℎ) = 119882(120574119905) 119875-as Thanks to 119882(120574119905) isF119905-measurable this relation holds true for all ℎ isin 119867
To finish the proof we also need the auxiliary lemma asfollows
Lemma 13 Let 120577 be a random variable defined over theclassical Wiener space (ΩF119879 119875) such that 120577(120591ℎ) = 120577 119875-asfor any ℎ isin 119867 Then 120577 = 119864120577 119875-as
Proof From Lemma 13 in Buckdahn and Li [10] we know forany 119860 isin B(R) 120593 isin 119871
2([0 119879]R119889
)
119864[1120577isin119860 expint
119879
0
120593 (119904) 119889119861 (119904)]
= 119864 [1120577isin119860] 119864 [expint
119879
0
120593 (119904) 119889119861 (119904)]
(36)
For any 120593 = sum119873
119894=11205931198941(119905
119894minus1119905119894] where 120593119894 isin R119889 for 0 le 119894 le 119873 and
119905119894119873
119894=0is a finite partition of [0 119879] from (36)
119864[1120577isin119860 exp(
119873
sum
119894=1
120593119894119861 (119905119894) minus 119861 (119905119894minus1))]
= 119864 [1120577isin119860] 119864[exp119873
sum
119894=1
120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]
(37)
Therefore for any nonnegative integer 119896119894
119864[1120577isin119860
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
= 119864 [1120577isin119860] 119864[
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
(38)
So for any polynomial function 119876 we have
119864 [1120577isin119860119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
= 119864 [1120577isin119860] 119864 [119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
(39)
Furthermore for any 119876 isin 119862119887(R119899) we still have (39)
Combining the arbitrariness of 119860 isin B(R) we obtain 120577 isindependent of (119861(1199051) minus 119861(1199050) 119861(119905119899) minus 119861(119905119899minus1)) for allpartition of [0 119879] Therefore 120577 is independent of F119879 whichimplies 120577 is independent of itself that is 119864120577 = 120577 119875-as
In Ji and Yang [14] they proved the following estimates
Mathematical Problems in Engineering 7
Lemma 14 Under the assumption (H) there exists someconstant 119862 gt 0 such that for any 119905 isin [0 119879] 120574119905 120574119905 isin Λ 119906(sdot) isin
U 119907(sdot) isin V
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905]le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119883
120574119905119906119907
(119904) minus 119883120574119905119906119907
(119904)10038161003816100381610038161003816
2
| F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
119864 [ sup119904isin[119905119879]
1003816100381610038161003816119884120574119905119906119907
(119904)10038161003816100381610038162+int
119879
119905
1003816100381610038161003816119885120574119905119906119907
(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119884120574119905119906119907
(119904) minus 119884120574119905119906119907
(119904)10038161003816100381610038161003816
2
+int
119879
119905
10038161003816100381610038161003816119885120574119905119906119907
(119904) minus 119885120574119905119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
(40)From the definition of119882(120574119905) and Lemma 14 we have the
following property
Lemma 15 There exists some constant 119862 gt 0 such that for all0 le 119905 le 119879 120574119905 120574119905 isin Λ
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817
(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171205741199051003817100381710038171003817)
(41)
Nowwe adopt Pengrsquos notion of stochastic backward semi-group (which was first introduced by Peng [9] to prove theDPP for stochastic control problems) to discuss a generalizedDPP for our SDG (17) (22) First we define the family ofbackward semigroups associated with FBSDE (17)
For given 119905 isin [0 119879] 120574119905 isin Λ a number 120575 isin (0 119879 minus 119905]admissible control processes 119906(sdot) isin U119905119905+120575 119907(sdot) isin V119905119905+120575 weset
119866120574119905119906119907
119904119905+120575[120578] =
120574119905119906119907
(119904) 119904 isin [119905 119905 + 120575] (42)
where 120578 isin 1198712(ΩF119905+120575
119905 119875R) (
120574119905119906119907
(119904) 119885120574119905119906119907
(119904))119905le119904le119905+120575
solves the following functional FBSDE on [119905 119905 + 120575]
119889120574119905119906119907
(119904)
= minus119891 (119904 119883120574119905119906119907
119904
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
120574119905119906119907
(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]
(43)Also we have
119866120574119905119906119907
119905119879[Φ (119883
120574119905119906119907
119879)] = 119866
120574119905119906119907
119905119905+120575[119884
120574119905119906119907
(119905 + 120575)] (44)Theorem 16 Suppose (H) holds true the lower value function119882(120574119905) satisfies the followingDPP for any 119905 isin [0 119879] 120574119905 isin Λ 120575 gt
0
119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (45)
The proof is given in the appendix
4 Viscosity Solutions of Path-DependentHJBI Equation
Now we study the following path-dependent PDEs
119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0
119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ
(46)
119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0
119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ
(47)
where
119867minus(120574119905119882119863119909119882119863119909119909119882)
= sup119906isin119880
inf119907isin119881
119867(120574119905119882119863119909119882119863119909119909119882119906 119907)
119867+(120574119905 119880119863119909119880119863119909119909119880)
= inf119907isin119881
sup119906isin119880
119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)
119867 (120574119905 119910 119901 119883 119906 119907)
=1
2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)
+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)
(48)
where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of
119889 times 119889 symmetric matrices)We will show that the value function119882(120574119905) (resp 119880(120574119905))
defined in (22) (resp (23)) is a viscosity solution of thecorresponding equation (46) (resp (47)) First we give thedefinition of viscosity solution for this kind of PDEs Formore information on viscosity solution the reader is referredto Crandall et al [22]
Definition 17 A real-valued Λ-continuous function 119882 isin
C(Λ) is called
(i) a viscosity subsolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ ge 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)
(ii) a viscosity supersolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ le 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)
(iii) a viscosity solution of (46) if it is both a viscosity sub-and supersolution of (46)
Theorem 18 Assume (H) holds the lower value function119882 isa viscosity solution of path-dependent HJBI Equation (46) onΛ the upper value function 119880 is a viscosity solution of (47)
8 Mathematical Problems in Engineering
First we prove some helpful lemmas For some fixed Γ isin
C12
119897119887(Λ) denote
119865 (120574119904 119910 119911 119906 119907)
= 119863119904Γ (120574119904) +1
2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))
+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)
+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)
(51)
where (120574119904 119910 119911 119906 119907) isin Λ timesR timesR119889times 119880 times 119881
Consider the following BSDE
minus 1198891198841119906119907
(119904)
= 119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198851119906119907
(119904) 119889119861 (119904)
1198841119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]
(52)
Lemma 19 For every 119904 isin [119905 119905 + 120575] one has the following
1198841119906119907
(119904) = 119866120574119905119906119907
119904119905+120575[Γ (119883
120574119905119906119907
119905+120575)] minus Γ (119883
120574119905119906119907
119904) 119875-as (53)
Proof Note119866120574119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] is defined as119866120574
119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] =
119884119906119907
(119904) through the following BSDE
minus 119889119884119906119907
(119904) = 119891 (119883120574119905119906119907
119904 119884
119906119907(119904) 119885
119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 119885119906119907
(119904) 119889119861 (119904)
119884119906119907
(119905 + 120575) = Γ (119883120574119905119906119907
119905+120575) 119904 isin [119905 119905 + 120575]
(54)
Using Itorsquos formula to Γ(119883120574119905119906119907
119904) we have
119889 (119884119906119907
(119904) minus Γ (119883120574119905119906119907
119904)) = 119889119884
1119906119907(119904) (55)
Combined with 119884119906119907
(119905 + 120575) minus Γ(119883120574119905119906119907
119905+120575) = 0 = 119884
1119906119907(119905 + 120575) we
get the desired result
Now consider the following BSDE
minus 1198891198842119906119907
(119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198852119906119907
(119904) 119889119861 (119904)
(56)
1198842119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)
Then we have the following lemma
Lemma 20 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816le 119862120575
32 119875-as (58)
where 119862 is independent of the control processes 119906 119907
Proof From Lemma 14 we know there exists some constant119862 gt 0 such that
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905] le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172) (59)
combined with
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]
le 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
10038161003816100381610038161003816100381610038161003816
2
| F119905]
+ 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)
10038161003816100381610038161003816100381610038161003816
2
| F119905]
(60)
we have
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)
From (52) and (54) using Lemma 8 set
1205851 = 1205852 = 0
119892 (119904 119910 119911) = 119865 (119883120574119905119906119907
119904 119910 119911 119906119904 119907119904)
1205931 (119904) = 0
1205932 (119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906119904 119907119904)
minus 119865 (119883120574119905119906119907
119904 119884
2119906119907(119904) 119885
2119906119907(119904) 119906119904 119907119904)
(62)
Denote 1205880(119903) = (1 + |119909|2)119903 due to 119887 120590 119891 are Lipishtiz and
that they are of linear growth Γ isin C12
119887 |1205932(119904)| le 1205880(|119883
120574119905119906119907
119904minus
120574119905(119905)|) we have
119864[int
119905+120575
119905
(100381610038161003816100381610038161198841119906119907
(119904) minus1198842119906119907
(119904)10038161003816100381610038161003816
2
+100381610038161003816100381610038161198851119906119907
(119904) minus1198852119906119907
(119904)10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864[int
119905+120575
119905
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) 119889119904 | F119905]
le 119862120575119864[ sup119904isin[119905119905+120575]
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) | F119905]
le 1198621205752
(63)
Mathematical Problems in Engineering 9
So100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816
=10038161003816100381610038161003816119864 [(119884
1119906119907(119904) minus 119884
2119906119907(119904)) | F119905]
10038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
119864 [int
119905+120575
119905
(119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904))
minus119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]
100381610038161003816100381610038161003816100381610038161003816
le 119862119864[int
119905+120575
119905
[1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
+100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816] 119889119904 | F119905]
le 119862119864[int
119905+120575
119905
1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) 119889119904 | F119905]
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
le 11986212057532
(64)
Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation
minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]
1198840 (119905 + 120575) = 0
(65)
where 1198650(120574119905 119910 119911) = sup119906isin119880
inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as
1198840 (119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) (66)
Proof First we define a function as follows
1198651 (120574119905 119910 119911 119906) = inf119907isin119881
119865 (120574119905 119910 119911 119906 119907)
(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880
(67)
Consider the following equation
minus1198891198843119906
(119904) = 1198651 (120574119905 1198843119906
(119904) 1198853119906
(119904) 119906 (119904)) 119889119904
minus1198853119906
(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]
1198843119906
(119905 + 120575) = 0
(68)
according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906
1198853119906
) solving (68) Also
1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as for every 119906 isin U119905119905+120575
(69)
In fact according to the definition of 1198651 and Lemma 7 wehave
1198843119906
(119905) le 1198842119906119907
(119905) 119875-as for any 119907 isin V119905119905+120575
for every 119906 isin U119905119905+120575
(70)
On the other side we have the existence of a measurablefunction 119907
4 R timesR119889
times 119880 rarr 119881 such that
1198651 (120574119905 119910 119911 119906)
= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906
(71)
Set 1199074(119904) = 1199074(119884
3119906(119904) 119885
3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know
1199074isin V119905119905+120575 Therefore (1198843119906
1198853119906
) = (11988421199061199074
11988521199061199074
) which isdue to the uniqueness of the solution of (68) In particular1198843119906
(119905) = 11988421199061199074
(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-119886119904 for every 119906 isin U119905119905+120575
Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880
1198651(120574119905 119910 119911 119906) wealso derive
1198840 (119905) = esssup119906isinU119905119905+120575
1198843119906
(119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as
(72)
Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575
32 119875-as
(73)
where the constant 119862 is independent of the control processes119906 119907
Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as
100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
2
le 119862120575
119864 [int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904] le 119862120575
(74)
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Definition 11 A nonanticipative strategy for Player I on[119905 119904] (119905 lt 119904 le 119879) is a mapping 120572 V119905119904 rarr U119905119904 suchthat for any F-stopping time 119878 Ω rarr [119905 119904] and any1199071 1199072 isin V119905119904 with 1199071 equiv 1199072 on [[119905 119878]] it holds that 120572(1199071) equiv
120572(1199072) on [[119905 119878]] Nonanticipative strategies for Player II on[119905 119904] 120573 U119905119904 rarr V119905119904 are defined similarly The set ofall nonanticipative strategies 120572 V119905119904 rarr U119905119904 for PlayerI on [119905 119904] is denoted by A119905119904 The set of all nonanticipativestrategies 120573 U119905119904 rarr V119905119904 for Player II on [119905 119904] is denoted byB119905119904 (Recall that [[119905 119878]] = (119903 120596) isin [0 119879] times Ω 119905 le 119903 le 119878(120596))
For given processes 119906(sdot) isin U119905119879 119907(sdot) isin V119905119879 initial data119905 isin [0 119879] 120574119905 isin Λ the cost functional is defined as follows
119869 (120574119905 119906 119907) = 119884120574119905119906119907
(119905) 120574119905 isin Λ (21)
where the process 119884120574119905119906119907 is defined by functional FBSDE (17)
For 120574119905 isin Λ the lower and the upper value functions of ourSDGs are defined as
119882(120574119905) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (22)
119880 (120574119905) = esssup120572isinA119905119879
essinf119907isinV119905119879
119869 (120574119905 120572 (119907) 119907) (23)
As we know the essential infimum and essential supre-mum on a family of random variables are still random vari-ables But by applying the method introduced by Buckdahnand Li [10] we get119882(120574119905) and 119880(120574119905) are deterministic
Proposition 12 For any 119905 isin [0 119879] 120574119905 isin Λ 119882(120574119905) is a deter-ministic function in the sense that119882(120574119905) = 119864[119882(120574119905)] 119875-as
Proof Let 119867 denote the Cameron-Martin space of all abso-lutely continuous elements ℎ isin Ω whose derivative ℎ belongsto 119871
2([0 119879]R119889
)For any ℎ isin 119867 we define the mapping 120591ℎ120596 = 120596 + ℎ 120596 isin
Ω It is easy to check that 120591ℎ Ω rarr Ω is a bijectionand its law is given by 119875 ∘ [120591ℎ]
minus1= expint119879
0ℎ(119904)119889119861(119904) minus
(12) int119879
0|ℎ(119904)|
2119889119904119875 For any fixed 119905 isin [0 119879] set 119867119905 = ℎ isin
119867 | ℎ(sdot) = ℎ(sdot and 119905) The proof can be separated into thefollowing four steps
(1) For all 119906 isin U119905119879 119907 isin V119905119879 ℎ isin 119867119905 119869(120574119905 119906 119907)(120591ℎ) =
119869(120574119905 119906(120591ℎ) 119907(120591ℎ)) 119875-asFirst we make the transformation for the functional SDE
as follows
119883120574119905119906119907
(119904)∘(120591ℎ)= (120574119905 (119905)+int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
+int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)
= 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906119907
119903(120591ℎ) 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int
119904
119905
119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119903
+ int
119904
119905
120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)
119903 119906 (120591ℎ) (119903)
119907 (120591ℎ) (119903)) 119889119861 (119903)
(24)
then from the uniqueness of the solution of the functionalSDE we get
119883120574119905119906119907
(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as(25)
Similarly using the transformation to the BSDE in (17) andcomparing the obtained equation with the BSDE obtainedfrom (17) by replacing the transformed control process119906(120591ℎ) 119907(120591ℎ) for 119906 119907 due to the uniqueness of the solution offunctional BSDE we obtain
119884120574119905119906119907
(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)
for any 119904 isin [119905 119879] 119875-as
119885120574119905119906119907
(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)
119889119904119889119875-ae on [0 119879] times Ω
(26)
Hence
119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)
(2) For 120573 isin B119905119879 ℎ isin 119867119905 let 120573ℎ(119906) = 120573(119906(120591minusℎ))(120591ℎ) 119906 isin
U119905119879 Then 120573ℎisin B119905119879
Obviously 120573ℎ U119905119879 rarr V119905119879 And it is nonanticipating
In fact given an F-stopping time 119878 Ω rarr [119905 119879] and 1199061 1199062 isin
U119905119879 with 1199061 equiv 1199062 on [[119905 119878]] Accordingly 1199061(120591minusℎ) equiv 1199062(120591minusℎ)
on [[119905 119878(120591minusℎ)]] Thus
120573ℎ(1199061) = 120573 (1199061 (120591minusℎ)) (120591ℎ) = 120573 (1199062 (120591minusℎ)) (120591ℎ)
= 120573ℎ(1199062) on [[119905 119878]]
(28)
(3) For any ℎ isin 119867119905 and 120573 isin B119905119879 we have
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)
In fact for convenience setting 119868(120574119905 120573) = esssup119906isinU119905119879
119869(120574119905 119906
120573(119906)) 120573 isin B119905119879 we know 119868(120574119905 120573) ge 119869(120574119905 119906 120573(119906)) Then
6 Mathematical Problems in Engineering
119868(120574119905 120573)(120591ℎ) ge 119869(120574119905 119906 120573(119906))(120591ℎ) 119875-as for all 119906 isin U119905119879Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)
From the definition of essential supremum for anyrandom variable 120585 which satisfies 120585 ge 119869(120574119905 119906 120573(119906))(120591ℎ) wehave 120585(120591minusℎ) ge 119869(120574119905 119906 120573(119906)) 119875-as for all 119906 isin U119905119879 So120585(120591minusℎ) ge 119868(120574119905 120573) 119875-as that is 120585 ge 119868(120574119905 120573)(120591ℎ) 119875-as Thus
119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
119875-as for any 119906 isin U119905119879
(31)
Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)
From above we get
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)
(4) Under the Girsanov transformation 120591ℎ 119882(120574119905) isinvariant that is
119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)
In fact we can prove essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) =
essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) 119875-as for all ℎ isin 119867119905 which issimilar to the above step From the above three steps for allℎ isin 119867119905 we get
119882(120574119905) (120591ℎ) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906))
= 119882 (120574119905) 119875-as(35)
Note that 119906(120591ℎ) | 119906(sdot) isin U119905119879 = U119905119879 and 120573ℎ
| 120573 isin
B119905119879 = B119905119879 have been used in the above equalities So forany ℎ isin 119867119905119882(120574119905)(120591ℎ) = 119882(120574119905) 119875-as Thanks to 119882(120574119905) isF119905-measurable this relation holds true for all ℎ isin 119867
To finish the proof we also need the auxiliary lemma asfollows
Lemma 13 Let 120577 be a random variable defined over theclassical Wiener space (ΩF119879 119875) such that 120577(120591ℎ) = 120577 119875-asfor any ℎ isin 119867 Then 120577 = 119864120577 119875-as
Proof From Lemma 13 in Buckdahn and Li [10] we know forany 119860 isin B(R) 120593 isin 119871
2([0 119879]R119889
)
119864[1120577isin119860 expint
119879
0
120593 (119904) 119889119861 (119904)]
= 119864 [1120577isin119860] 119864 [expint
119879
0
120593 (119904) 119889119861 (119904)]
(36)
For any 120593 = sum119873
119894=11205931198941(119905
119894minus1119905119894] where 120593119894 isin R119889 for 0 le 119894 le 119873 and
119905119894119873
119894=0is a finite partition of [0 119879] from (36)
119864[1120577isin119860 exp(
119873
sum
119894=1
120593119894119861 (119905119894) minus 119861 (119905119894minus1))]
= 119864 [1120577isin119860] 119864[exp119873
sum
119894=1
120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]
(37)
Therefore for any nonnegative integer 119896119894
119864[1120577isin119860
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
= 119864 [1120577isin119860] 119864[
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
(38)
So for any polynomial function 119876 we have
119864 [1120577isin119860119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
= 119864 [1120577isin119860] 119864 [119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
(39)
Furthermore for any 119876 isin 119862119887(R119899) we still have (39)
Combining the arbitrariness of 119860 isin B(R) we obtain 120577 isindependent of (119861(1199051) minus 119861(1199050) 119861(119905119899) minus 119861(119905119899minus1)) for allpartition of [0 119879] Therefore 120577 is independent of F119879 whichimplies 120577 is independent of itself that is 119864120577 = 120577 119875-as
In Ji and Yang [14] they proved the following estimates
Mathematical Problems in Engineering 7
Lemma 14 Under the assumption (H) there exists someconstant 119862 gt 0 such that for any 119905 isin [0 119879] 120574119905 120574119905 isin Λ 119906(sdot) isin
U 119907(sdot) isin V
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905]le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119883
120574119905119906119907
(119904) minus 119883120574119905119906119907
(119904)10038161003816100381610038161003816
2
| F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
119864 [ sup119904isin[119905119879]
1003816100381610038161003816119884120574119905119906119907
(119904)10038161003816100381610038162+int
119879
119905
1003816100381610038161003816119885120574119905119906119907
(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119884120574119905119906119907
(119904) minus 119884120574119905119906119907
(119904)10038161003816100381610038161003816
2
+int
119879
119905
10038161003816100381610038161003816119885120574119905119906119907
(119904) minus 119885120574119905119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
(40)From the definition of119882(120574119905) and Lemma 14 we have the
following property
Lemma 15 There exists some constant 119862 gt 0 such that for all0 le 119905 le 119879 120574119905 120574119905 isin Λ
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817
(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171205741199051003817100381710038171003817)
(41)
Nowwe adopt Pengrsquos notion of stochastic backward semi-group (which was first introduced by Peng [9] to prove theDPP for stochastic control problems) to discuss a generalizedDPP for our SDG (17) (22) First we define the family ofbackward semigroups associated with FBSDE (17)
For given 119905 isin [0 119879] 120574119905 isin Λ a number 120575 isin (0 119879 minus 119905]admissible control processes 119906(sdot) isin U119905119905+120575 119907(sdot) isin V119905119905+120575 weset
119866120574119905119906119907
119904119905+120575[120578] =
120574119905119906119907
(119904) 119904 isin [119905 119905 + 120575] (42)
where 120578 isin 1198712(ΩF119905+120575
119905 119875R) (
120574119905119906119907
(119904) 119885120574119905119906119907
(119904))119905le119904le119905+120575
solves the following functional FBSDE on [119905 119905 + 120575]
119889120574119905119906119907
(119904)
= minus119891 (119904 119883120574119905119906119907
119904
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
120574119905119906119907
(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]
(43)Also we have
119866120574119905119906119907
119905119879[Φ (119883
120574119905119906119907
119879)] = 119866
120574119905119906119907
119905119905+120575[119884
120574119905119906119907
(119905 + 120575)] (44)Theorem 16 Suppose (H) holds true the lower value function119882(120574119905) satisfies the followingDPP for any 119905 isin [0 119879] 120574119905 isin Λ 120575 gt
0
119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (45)
The proof is given in the appendix
4 Viscosity Solutions of Path-DependentHJBI Equation
Now we study the following path-dependent PDEs
119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0
119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ
(46)
119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0
119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ
(47)
where
119867minus(120574119905119882119863119909119882119863119909119909119882)
= sup119906isin119880
inf119907isin119881
119867(120574119905119882119863119909119882119863119909119909119882119906 119907)
119867+(120574119905 119880119863119909119880119863119909119909119880)
= inf119907isin119881
sup119906isin119880
119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)
119867 (120574119905 119910 119901 119883 119906 119907)
=1
2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)
+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)
(48)
where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of
119889 times 119889 symmetric matrices)We will show that the value function119882(120574119905) (resp 119880(120574119905))
defined in (22) (resp (23)) is a viscosity solution of thecorresponding equation (46) (resp (47)) First we give thedefinition of viscosity solution for this kind of PDEs Formore information on viscosity solution the reader is referredto Crandall et al [22]
Definition 17 A real-valued Λ-continuous function 119882 isin
C(Λ) is called
(i) a viscosity subsolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ ge 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)
(ii) a viscosity supersolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ le 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)
(iii) a viscosity solution of (46) if it is both a viscosity sub-and supersolution of (46)
Theorem 18 Assume (H) holds the lower value function119882 isa viscosity solution of path-dependent HJBI Equation (46) onΛ the upper value function 119880 is a viscosity solution of (47)
8 Mathematical Problems in Engineering
First we prove some helpful lemmas For some fixed Γ isin
C12
119897119887(Λ) denote
119865 (120574119904 119910 119911 119906 119907)
= 119863119904Γ (120574119904) +1
2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))
+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)
+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)
(51)
where (120574119904 119910 119911 119906 119907) isin Λ timesR timesR119889times 119880 times 119881
Consider the following BSDE
minus 1198891198841119906119907
(119904)
= 119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198851119906119907
(119904) 119889119861 (119904)
1198841119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]
(52)
Lemma 19 For every 119904 isin [119905 119905 + 120575] one has the following
1198841119906119907
(119904) = 119866120574119905119906119907
119904119905+120575[Γ (119883
120574119905119906119907
119905+120575)] minus Γ (119883
120574119905119906119907
119904) 119875-as (53)
Proof Note119866120574119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] is defined as119866120574
119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] =
119884119906119907
(119904) through the following BSDE
minus 119889119884119906119907
(119904) = 119891 (119883120574119905119906119907
119904 119884
119906119907(119904) 119885
119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 119885119906119907
(119904) 119889119861 (119904)
119884119906119907
(119905 + 120575) = Γ (119883120574119905119906119907
119905+120575) 119904 isin [119905 119905 + 120575]
(54)
Using Itorsquos formula to Γ(119883120574119905119906119907
119904) we have
119889 (119884119906119907
(119904) minus Γ (119883120574119905119906119907
119904)) = 119889119884
1119906119907(119904) (55)
Combined with 119884119906119907
(119905 + 120575) minus Γ(119883120574119905119906119907
119905+120575) = 0 = 119884
1119906119907(119905 + 120575) we
get the desired result
Now consider the following BSDE
minus 1198891198842119906119907
(119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198852119906119907
(119904) 119889119861 (119904)
(56)
1198842119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)
Then we have the following lemma
Lemma 20 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816le 119862120575
32 119875-as (58)
where 119862 is independent of the control processes 119906 119907
Proof From Lemma 14 we know there exists some constant119862 gt 0 such that
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905] le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172) (59)
combined with
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]
le 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
10038161003816100381610038161003816100381610038161003816
2
| F119905]
+ 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)
10038161003816100381610038161003816100381610038161003816
2
| F119905]
(60)
we have
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)
From (52) and (54) using Lemma 8 set
1205851 = 1205852 = 0
119892 (119904 119910 119911) = 119865 (119883120574119905119906119907
119904 119910 119911 119906119904 119907119904)
1205931 (119904) = 0
1205932 (119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906119904 119907119904)
minus 119865 (119883120574119905119906119907
119904 119884
2119906119907(119904) 119885
2119906119907(119904) 119906119904 119907119904)
(62)
Denote 1205880(119903) = (1 + |119909|2)119903 due to 119887 120590 119891 are Lipishtiz and
that they are of linear growth Γ isin C12
119887 |1205932(119904)| le 1205880(|119883
120574119905119906119907
119904minus
120574119905(119905)|) we have
119864[int
119905+120575
119905
(100381610038161003816100381610038161198841119906119907
(119904) minus1198842119906119907
(119904)10038161003816100381610038161003816
2
+100381610038161003816100381610038161198851119906119907
(119904) minus1198852119906119907
(119904)10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864[int
119905+120575
119905
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) 119889119904 | F119905]
le 119862120575119864[ sup119904isin[119905119905+120575]
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) | F119905]
le 1198621205752
(63)
Mathematical Problems in Engineering 9
So100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816
=10038161003816100381610038161003816119864 [(119884
1119906119907(119904) minus 119884
2119906119907(119904)) | F119905]
10038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
119864 [int
119905+120575
119905
(119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904))
minus119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]
100381610038161003816100381610038161003816100381610038161003816
le 119862119864[int
119905+120575
119905
[1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
+100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816] 119889119904 | F119905]
le 119862119864[int
119905+120575
119905
1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) 119889119904 | F119905]
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
le 11986212057532
(64)
Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation
minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]
1198840 (119905 + 120575) = 0
(65)
where 1198650(120574119905 119910 119911) = sup119906isin119880
inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as
1198840 (119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) (66)
Proof First we define a function as follows
1198651 (120574119905 119910 119911 119906) = inf119907isin119881
119865 (120574119905 119910 119911 119906 119907)
(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880
(67)
Consider the following equation
minus1198891198843119906
(119904) = 1198651 (120574119905 1198843119906
(119904) 1198853119906
(119904) 119906 (119904)) 119889119904
minus1198853119906
(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]
1198843119906
(119905 + 120575) = 0
(68)
according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906
1198853119906
) solving (68) Also
1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as for every 119906 isin U119905119905+120575
(69)
In fact according to the definition of 1198651 and Lemma 7 wehave
1198843119906
(119905) le 1198842119906119907
(119905) 119875-as for any 119907 isin V119905119905+120575
for every 119906 isin U119905119905+120575
(70)
On the other side we have the existence of a measurablefunction 119907
4 R timesR119889
times 119880 rarr 119881 such that
1198651 (120574119905 119910 119911 119906)
= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906
(71)
Set 1199074(119904) = 1199074(119884
3119906(119904) 119885
3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know
1199074isin V119905119905+120575 Therefore (1198843119906
1198853119906
) = (11988421199061199074
11988521199061199074
) which isdue to the uniqueness of the solution of (68) In particular1198843119906
(119905) = 11988421199061199074
(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-119886119904 for every 119906 isin U119905119905+120575
Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880
1198651(120574119905 119910 119911 119906) wealso derive
1198840 (119905) = esssup119906isinU119905119905+120575
1198843119906
(119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as
(72)
Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575
32 119875-as
(73)
where the constant 119862 is independent of the control processes119906 119907
Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as
100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
2
le 119862120575
119864 [int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904] le 119862120575
(74)
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
119868(120574119905 120573)(120591ℎ) ge 119869(120574119905 119906 120573(119906))(120591ℎ) 119875-as for all 119906 isin U119905119879Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)
From the definition of essential supremum for anyrandom variable 120585 which satisfies 120585 ge 119869(120574119905 119906 120573(119906))(120591ℎ) wehave 120585(120591minusℎ) ge 119869(120574119905 119906 120573(119906)) 119875-as for all 119906 isin U119905119879 So120585(120591minusℎ) ge 119868(120574119905 120573) 119875-as that is 120585 ge 119868(120574119905 120573)(120591ℎ) 119875-as Thus
119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
119875-as for any 119906 isin U119905119879
(31)
Therefore
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
ge esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)
From above we get
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)
(4) Under the Girsanov transformation 120591ℎ 119882(120574119905) isinvariant that is
119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)
In fact we can prove essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) =
essinf120573isinB119905119879
119868(120574119905 120573)(120591ℎ) 119875-as for all ℎ isin 119867119905 which issimilar to the above step From the above three steps for allℎ isin 119867119905 we get
119882(120574119905) (120591ℎ) = essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) (120591ℎ)
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))
= essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906))
= 119882 (120574119905) 119875-as(35)
Note that 119906(120591ℎ) | 119906(sdot) isin U119905119879 = U119905119879 and 120573ℎ
| 120573 isin
B119905119879 = B119905119879 have been used in the above equalities So forany ℎ isin 119867119905119882(120574119905)(120591ℎ) = 119882(120574119905) 119875-as Thanks to 119882(120574119905) isF119905-measurable this relation holds true for all ℎ isin 119867
To finish the proof we also need the auxiliary lemma asfollows
Lemma 13 Let 120577 be a random variable defined over theclassical Wiener space (ΩF119879 119875) such that 120577(120591ℎ) = 120577 119875-asfor any ℎ isin 119867 Then 120577 = 119864120577 119875-as
Proof From Lemma 13 in Buckdahn and Li [10] we know forany 119860 isin B(R) 120593 isin 119871
2([0 119879]R119889
)
119864[1120577isin119860 expint
119879
0
120593 (119904) 119889119861 (119904)]
= 119864 [1120577isin119860] 119864 [expint
119879
0
120593 (119904) 119889119861 (119904)]
(36)
For any 120593 = sum119873
119894=11205931198941(119905
119894minus1119905119894] where 120593119894 isin R119889 for 0 le 119894 le 119873 and
119905119894119873
119894=0is a finite partition of [0 119879] from (36)
119864[1120577isin119860 exp(
119873
sum
119894=1
120593119894119861 (119905119894) minus 119861 (119905119894minus1))]
= 119864 [1120577isin119860] 119864[exp119873
sum
119894=1
120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]
(37)
Therefore for any nonnegative integer 119896119894
119864[1120577isin119860
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
= 119864 [1120577isin119860] 119864[
119873
prod
119894=1
(119861 (119905119894) minus 119861 (119905119894minus1))119896119894
]
(38)
So for any polynomial function 119876 we have
119864 [1120577isin119860119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
= 119864 [1120577isin119860] 119864 [119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]
(39)
Furthermore for any 119876 isin 119862119887(R119899) we still have (39)
Combining the arbitrariness of 119860 isin B(R) we obtain 120577 isindependent of (119861(1199051) minus 119861(1199050) 119861(119905119899) minus 119861(119905119899minus1)) for allpartition of [0 119879] Therefore 120577 is independent of F119879 whichimplies 120577 is independent of itself that is 119864120577 = 120577 119875-as
In Ji and Yang [14] they proved the following estimates
Mathematical Problems in Engineering 7
Lemma 14 Under the assumption (H) there exists someconstant 119862 gt 0 such that for any 119905 isin [0 119879] 120574119905 120574119905 isin Λ 119906(sdot) isin
U 119907(sdot) isin V
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905]le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119883
120574119905119906119907
(119904) minus 119883120574119905119906119907
(119904)10038161003816100381610038161003816
2
| F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
119864 [ sup119904isin[119905119879]
1003816100381610038161003816119884120574119905119906119907
(119904)10038161003816100381610038162+int
119879
119905
1003816100381610038161003816119885120574119905119906119907
(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119884120574119905119906119907
(119904) minus 119884120574119905119906119907
(119904)10038161003816100381610038161003816
2
+int
119879
119905
10038161003816100381610038161003816119885120574119905119906119907
(119904) minus 119885120574119905119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
(40)From the definition of119882(120574119905) and Lemma 14 we have the
following property
Lemma 15 There exists some constant 119862 gt 0 such that for all0 le 119905 le 119879 120574119905 120574119905 isin Λ
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817
(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171205741199051003817100381710038171003817)
(41)
Nowwe adopt Pengrsquos notion of stochastic backward semi-group (which was first introduced by Peng [9] to prove theDPP for stochastic control problems) to discuss a generalizedDPP for our SDG (17) (22) First we define the family ofbackward semigroups associated with FBSDE (17)
For given 119905 isin [0 119879] 120574119905 isin Λ a number 120575 isin (0 119879 minus 119905]admissible control processes 119906(sdot) isin U119905119905+120575 119907(sdot) isin V119905119905+120575 weset
119866120574119905119906119907
119904119905+120575[120578] =
120574119905119906119907
(119904) 119904 isin [119905 119905 + 120575] (42)
where 120578 isin 1198712(ΩF119905+120575
119905 119875R) (
120574119905119906119907
(119904) 119885120574119905119906119907
(119904))119905le119904le119905+120575
solves the following functional FBSDE on [119905 119905 + 120575]
119889120574119905119906119907
(119904)
= minus119891 (119904 119883120574119905119906119907
119904
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
120574119905119906119907
(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]
(43)Also we have
119866120574119905119906119907
119905119879[Φ (119883
120574119905119906119907
119879)] = 119866
120574119905119906119907
119905119905+120575[119884
120574119905119906119907
(119905 + 120575)] (44)Theorem 16 Suppose (H) holds true the lower value function119882(120574119905) satisfies the followingDPP for any 119905 isin [0 119879] 120574119905 isin Λ 120575 gt
0
119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (45)
The proof is given in the appendix
4 Viscosity Solutions of Path-DependentHJBI Equation
Now we study the following path-dependent PDEs
119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0
119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ
(46)
119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0
119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ
(47)
where
119867minus(120574119905119882119863119909119882119863119909119909119882)
= sup119906isin119880
inf119907isin119881
119867(120574119905119882119863119909119882119863119909119909119882119906 119907)
119867+(120574119905 119880119863119909119880119863119909119909119880)
= inf119907isin119881
sup119906isin119880
119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)
119867 (120574119905 119910 119901 119883 119906 119907)
=1
2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)
+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)
(48)
where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of
119889 times 119889 symmetric matrices)We will show that the value function119882(120574119905) (resp 119880(120574119905))
defined in (22) (resp (23)) is a viscosity solution of thecorresponding equation (46) (resp (47)) First we give thedefinition of viscosity solution for this kind of PDEs Formore information on viscosity solution the reader is referredto Crandall et al [22]
Definition 17 A real-valued Λ-continuous function 119882 isin
C(Λ) is called
(i) a viscosity subsolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ ge 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)
(ii) a viscosity supersolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ le 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)
(iii) a viscosity solution of (46) if it is both a viscosity sub-and supersolution of (46)
Theorem 18 Assume (H) holds the lower value function119882 isa viscosity solution of path-dependent HJBI Equation (46) onΛ the upper value function 119880 is a viscosity solution of (47)
8 Mathematical Problems in Engineering
First we prove some helpful lemmas For some fixed Γ isin
C12
119897119887(Λ) denote
119865 (120574119904 119910 119911 119906 119907)
= 119863119904Γ (120574119904) +1
2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))
+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)
+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)
(51)
where (120574119904 119910 119911 119906 119907) isin Λ timesR timesR119889times 119880 times 119881
Consider the following BSDE
minus 1198891198841119906119907
(119904)
= 119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198851119906119907
(119904) 119889119861 (119904)
1198841119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]
(52)
Lemma 19 For every 119904 isin [119905 119905 + 120575] one has the following
1198841119906119907
(119904) = 119866120574119905119906119907
119904119905+120575[Γ (119883
120574119905119906119907
119905+120575)] minus Γ (119883
120574119905119906119907
119904) 119875-as (53)
Proof Note119866120574119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] is defined as119866120574
119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] =
119884119906119907
(119904) through the following BSDE
minus 119889119884119906119907
(119904) = 119891 (119883120574119905119906119907
119904 119884
119906119907(119904) 119885
119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 119885119906119907
(119904) 119889119861 (119904)
119884119906119907
(119905 + 120575) = Γ (119883120574119905119906119907
119905+120575) 119904 isin [119905 119905 + 120575]
(54)
Using Itorsquos formula to Γ(119883120574119905119906119907
119904) we have
119889 (119884119906119907
(119904) minus Γ (119883120574119905119906119907
119904)) = 119889119884
1119906119907(119904) (55)
Combined with 119884119906119907
(119905 + 120575) minus Γ(119883120574119905119906119907
119905+120575) = 0 = 119884
1119906119907(119905 + 120575) we
get the desired result
Now consider the following BSDE
minus 1198891198842119906119907
(119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198852119906119907
(119904) 119889119861 (119904)
(56)
1198842119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)
Then we have the following lemma
Lemma 20 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816le 119862120575
32 119875-as (58)
where 119862 is independent of the control processes 119906 119907
Proof From Lemma 14 we know there exists some constant119862 gt 0 such that
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905] le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172) (59)
combined with
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]
le 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
10038161003816100381610038161003816100381610038161003816
2
| F119905]
+ 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)
10038161003816100381610038161003816100381610038161003816
2
| F119905]
(60)
we have
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)
From (52) and (54) using Lemma 8 set
1205851 = 1205852 = 0
119892 (119904 119910 119911) = 119865 (119883120574119905119906119907
119904 119910 119911 119906119904 119907119904)
1205931 (119904) = 0
1205932 (119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906119904 119907119904)
minus 119865 (119883120574119905119906119907
119904 119884
2119906119907(119904) 119885
2119906119907(119904) 119906119904 119907119904)
(62)
Denote 1205880(119903) = (1 + |119909|2)119903 due to 119887 120590 119891 are Lipishtiz and
that they are of linear growth Γ isin C12
119887 |1205932(119904)| le 1205880(|119883
120574119905119906119907
119904minus
120574119905(119905)|) we have
119864[int
119905+120575
119905
(100381610038161003816100381610038161198841119906119907
(119904) minus1198842119906119907
(119904)10038161003816100381610038161003816
2
+100381610038161003816100381610038161198851119906119907
(119904) minus1198852119906119907
(119904)10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864[int
119905+120575
119905
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) 119889119904 | F119905]
le 119862120575119864[ sup119904isin[119905119905+120575]
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) | F119905]
le 1198621205752
(63)
Mathematical Problems in Engineering 9
So100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816
=10038161003816100381610038161003816119864 [(119884
1119906119907(119904) minus 119884
2119906119907(119904)) | F119905]
10038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
119864 [int
119905+120575
119905
(119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904))
minus119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]
100381610038161003816100381610038161003816100381610038161003816
le 119862119864[int
119905+120575
119905
[1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
+100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816] 119889119904 | F119905]
le 119862119864[int
119905+120575
119905
1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) 119889119904 | F119905]
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
le 11986212057532
(64)
Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation
minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]
1198840 (119905 + 120575) = 0
(65)
where 1198650(120574119905 119910 119911) = sup119906isin119880
inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as
1198840 (119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) (66)
Proof First we define a function as follows
1198651 (120574119905 119910 119911 119906) = inf119907isin119881
119865 (120574119905 119910 119911 119906 119907)
(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880
(67)
Consider the following equation
minus1198891198843119906
(119904) = 1198651 (120574119905 1198843119906
(119904) 1198853119906
(119904) 119906 (119904)) 119889119904
minus1198853119906
(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]
1198843119906
(119905 + 120575) = 0
(68)
according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906
1198853119906
) solving (68) Also
1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as for every 119906 isin U119905119905+120575
(69)
In fact according to the definition of 1198651 and Lemma 7 wehave
1198843119906
(119905) le 1198842119906119907
(119905) 119875-as for any 119907 isin V119905119905+120575
for every 119906 isin U119905119905+120575
(70)
On the other side we have the existence of a measurablefunction 119907
4 R timesR119889
times 119880 rarr 119881 such that
1198651 (120574119905 119910 119911 119906)
= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906
(71)
Set 1199074(119904) = 1199074(119884
3119906(119904) 119885
3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know
1199074isin V119905119905+120575 Therefore (1198843119906
1198853119906
) = (11988421199061199074
11988521199061199074
) which isdue to the uniqueness of the solution of (68) In particular1198843119906
(119905) = 11988421199061199074
(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-119886119904 for every 119906 isin U119905119905+120575
Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880
1198651(120574119905 119910 119911 119906) wealso derive
1198840 (119905) = esssup119906isinU119905119905+120575
1198843119906
(119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as
(72)
Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575
32 119875-as
(73)
where the constant 119862 is independent of the control processes119906 119907
Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as
100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
2
le 119862120575
119864 [int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904] le 119862120575
(74)
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Lemma 14 Under the assumption (H) there exists someconstant 119862 gt 0 such that for any 119905 isin [0 119879] 120574119905 120574119905 isin Λ 119906(sdot) isin
U 119907(sdot) isin V
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905]le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119883
120574119905119906119907
(119904) minus 119883120574119905119906119907
(119904)10038161003816100381610038161003816
2
| F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
119864 [ sup119904isin[119905119879]
1003816100381610038161003816119884120574119905119906119907
(119904)10038161003816100381610038162+int
119879
119905
1003816100381610038161003816119885120574119905119906119907
(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+
100381710038171003817100381712057411990510038171003817100381710038172)
119864 [ sup119904isin[119905119879]
10038161003816100381610038161003816119884120574119905119906119907
(119904) minus 119884120574119905119906119907
(119904)10038161003816100381610038161003816
2
+int
119879
119905
10038161003816100381610038161003816119885120574119905119906119907
(119904) minus 119885120574119905119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574
119905
10038171003817100381710038172
(40)From the definition of119882(120574119905) and Lemma 14 we have the
following property
Lemma 15 There exists some constant 119862 gt 0 such that for all0 le 119905 le 119879 120574119905 120574119905 isin Λ
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817
(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +
10038171003817100381710038171205741199051003817100381710038171003817)
(41)
Nowwe adopt Pengrsquos notion of stochastic backward semi-group (which was first introduced by Peng [9] to prove theDPP for stochastic control problems) to discuss a generalizedDPP for our SDG (17) (22) First we define the family ofbackward semigroups associated with FBSDE (17)
For given 119905 isin [0 119879] 120574119905 isin Λ a number 120575 isin (0 119879 minus 119905]admissible control processes 119906(sdot) isin U119905119905+120575 119907(sdot) isin V119905119905+120575 weset
119866120574119905119906119907
119904119905+120575[120578] =
120574119905119906119907
(119904) 119904 isin [119905 119905 + 120575] (42)
where 120578 isin 1198712(ΩF119905+120575
119905 119875R) (
120574119905119906119907
(119904) 119885120574119905119906119907
(119904))119905le119904le119905+120575
solves the following functional FBSDE on [119905 119905 + 120575]
119889120574119905119906119907
(119904)
= minus119891 (119904 119883120574119905119906119907
119904
120574119905119906119907
(119904) 119885120574119905119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
+ 119885120574119905119906119907
(119904) 119889119861 (119904)
120574119905119906119907
(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]
(43)Also we have
119866120574119905119906119907
119905119879[Φ (119883
120574119905119906119907
119879)] = 119866
120574119905119906119907
119905119905+120575[119884
120574119905119906119907
(119905 + 120575)] (44)Theorem 16 Suppose (H) holds true the lower value function119882(120574119905) satisfies the followingDPP for any 119905 isin [0 119879] 120574119905 isin Λ 120575 gt
0
119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (45)
The proof is given in the appendix
4 Viscosity Solutions of Path-DependentHJBI Equation
Now we study the following path-dependent PDEs
119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0
119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ
(46)
119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0
119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ
(47)
where
119867minus(120574119905119882119863119909119882119863119909119909119882)
= sup119906isin119880
inf119907isin119881
119867(120574119905119882119863119909119882119863119909119909119882119906 119907)
119867+(120574119905 119880119863119909119880119863119909119909119880)
= inf119907isin119881
sup119906isin119880
119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)
119867 (120574119905 119910 119901 119883 119906 119907)
=1
2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)
+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)
(48)
where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of
119889 times 119889 symmetric matrices)We will show that the value function119882(120574119905) (resp 119880(120574119905))
defined in (22) (resp (23)) is a viscosity solution of thecorresponding equation (46) (resp (47)) First we give thedefinition of viscosity solution for this kind of PDEs Formore information on viscosity solution the reader is referredto Crandall et al [22]
Definition 17 A real-valued Λ-continuous function 119882 isin
C(Λ) is called
(i) a viscosity subsolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ ge 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)
(ii) a viscosity supersolution of (46) if for any 120575 gt 0 Γ isin
C12
119897119887(Λ) 120574119905 isin Λ satisfying Γ le 119882 on ⋃
0le119904le120575Λ 119905+119904 and
Γ(120574119905) = 119882(120574119905) one has
119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)
(iii) a viscosity solution of (46) if it is both a viscosity sub-and supersolution of (46)
Theorem 18 Assume (H) holds the lower value function119882 isa viscosity solution of path-dependent HJBI Equation (46) onΛ the upper value function 119880 is a viscosity solution of (47)
8 Mathematical Problems in Engineering
First we prove some helpful lemmas For some fixed Γ isin
C12
119897119887(Λ) denote
119865 (120574119904 119910 119911 119906 119907)
= 119863119904Γ (120574119904) +1
2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))
+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)
+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)
(51)
where (120574119904 119910 119911 119906 119907) isin Λ timesR timesR119889times 119880 times 119881
Consider the following BSDE
minus 1198891198841119906119907
(119904)
= 119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198851119906119907
(119904) 119889119861 (119904)
1198841119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]
(52)
Lemma 19 For every 119904 isin [119905 119905 + 120575] one has the following
1198841119906119907
(119904) = 119866120574119905119906119907
119904119905+120575[Γ (119883
120574119905119906119907
119905+120575)] minus Γ (119883
120574119905119906119907
119904) 119875-as (53)
Proof Note119866120574119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] is defined as119866120574
119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] =
119884119906119907
(119904) through the following BSDE
minus 119889119884119906119907
(119904) = 119891 (119883120574119905119906119907
119904 119884
119906119907(119904) 119885
119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 119885119906119907
(119904) 119889119861 (119904)
119884119906119907
(119905 + 120575) = Γ (119883120574119905119906119907
119905+120575) 119904 isin [119905 119905 + 120575]
(54)
Using Itorsquos formula to Γ(119883120574119905119906119907
119904) we have
119889 (119884119906119907
(119904) minus Γ (119883120574119905119906119907
119904)) = 119889119884
1119906119907(119904) (55)
Combined with 119884119906119907
(119905 + 120575) minus Γ(119883120574119905119906119907
119905+120575) = 0 = 119884
1119906119907(119905 + 120575) we
get the desired result
Now consider the following BSDE
minus 1198891198842119906119907
(119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198852119906119907
(119904) 119889119861 (119904)
(56)
1198842119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)
Then we have the following lemma
Lemma 20 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816le 119862120575
32 119875-as (58)
where 119862 is independent of the control processes 119906 119907
Proof From Lemma 14 we know there exists some constant119862 gt 0 such that
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905] le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172) (59)
combined with
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]
le 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
10038161003816100381610038161003816100381610038161003816
2
| F119905]
+ 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)
10038161003816100381610038161003816100381610038161003816
2
| F119905]
(60)
we have
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)
From (52) and (54) using Lemma 8 set
1205851 = 1205852 = 0
119892 (119904 119910 119911) = 119865 (119883120574119905119906119907
119904 119910 119911 119906119904 119907119904)
1205931 (119904) = 0
1205932 (119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906119904 119907119904)
minus 119865 (119883120574119905119906119907
119904 119884
2119906119907(119904) 119885
2119906119907(119904) 119906119904 119907119904)
(62)
Denote 1205880(119903) = (1 + |119909|2)119903 due to 119887 120590 119891 are Lipishtiz and
that they are of linear growth Γ isin C12
119887 |1205932(119904)| le 1205880(|119883
120574119905119906119907
119904minus
120574119905(119905)|) we have
119864[int
119905+120575
119905
(100381610038161003816100381610038161198841119906119907
(119904) minus1198842119906119907
(119904)10038161003816100381610038161003816
2
+100381610038161003816100381610038161198851119906119907
(119904) minus1198852119906119907
(119904)10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864[int
119905+120575
119905
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) 119889119904 | F119905]
le 119862120575119864[ sup119904isin[119905119905+120575]
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) | F119905]
le 1198621205752
(63)
Mathematical Problems in Engineering 9
So100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816
=10038161003816100381610038161003816119864 [(119884
1119906119907(119904) minus 119884
2119906119907(119904)) | F119905]
10038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
119864 [int
119905+120575
119905
(119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904))
minus119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]
100381610038161003816100381610038161003816100381610038161003816
le 119862119864[int
119905+120575
119905
[1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
+100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816] 119889119904 | F119905]
le 119862119864[int
119905+120575
119905
1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) 119889119904 | F119905]
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
le 11986212057532
(64)
Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation
minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]
1198840 (119905 + 120575) = 0
(65)
where 1198650(120574119905 119910 119911) = sup119906isin119880
inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as
1198840 (119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) (66)
Proof First we define a function as follows
1198651 (120574119905 119910 119911 119906) = inf119907isin119881
119865 (120574119905 119910 119911 119906 119907)
(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880
(67)
Consider the following equation
minus1198891198843119906
(119904) = 1198651 (120574119905 1198843119906
(119904) 1198853119906
(119904) 119906 (119904)) 119889119904
minus1198853119906
(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]
1198843119906
(119905 + 120575) = 0
(68)
according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906
1198853119906
) solving (68) Also
1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as for every 119906 isin U119905119905+120575
(69)
In fact according to the definition of 1198651 and Lemma 7 wehave
1198843119906
(119905) le 1198842119906119907
(119905) 119875-as for any 119907 isin V119905119905+120575
for every 119906 isin U119905119905+120575
(70)
On the other side we have the existence of a measurablefunction 119907
4 R timesR119889
times 119880 rarr 119881 such that
1198651 (120574119905 119910 119911 119906)
= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906
(71)
Set 1199074(119904) = 1199074(119884
3119906(119904) 119885
3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know
1199074isin V119905119905+120575 Therefore (1198843119906
1198853119906
) = (11988421199061199074
11988521199061199074
) which isdue to the uniqueness of the solution of (68) In particular1198843119906
(119905) = 11988421199061199074
(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-119886119904 for every 119906 isin U119905119905+120575
Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880
1198651(120574119905 119910 119911 119906) wealso derive
1198840 (119905) = esssup119906isinU119905119905+120575
1198843119906
(119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as
(72)
Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575
32 119875-as
(73)
where the constant 119862 is independent of the control processes119906 119907
Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as
100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
2
le 119862120575
119864 [int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904] le 119862120575
(74)
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
First we prove some helpful lemmas For some fixed Γ isin
C12
119897119887(Λ) denote
119865 (120574119904 119910 119911 119906 119907)
= 119863119904Γ (120574119904) +1
2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))
+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)
+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)
(51)
where (120574119904 119910 119911 119906 119907) isin Λ timesR timesR119889times 119880 times 119881
Consider the following BSDE
minus 1198891198841119906119907
(119904)
= 119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198851119906119907
(119904) 119889119861 (119904)
1198841119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]
(52)
Lemma 19 For every 119904 isin [119905 119905 + 120575] one has the following
1198841119906119907
(119904) = 119866120574119905119906119907
119904119905+120575[Γ (119883
120574119905119906119907
119905+120575)] minus Γ (119883
120574119905119906119907
119904) 119875-as (53)
Proof Note119866120574119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] is defined as119866120574
119905119906119907
119904119905+120575[Γ(119883
120574119905119906119907
119905+120575)] =
119884119906119907
(119904) through the following BSDE
minus 119889119884119906119907
(119904) = 119891 (119883120574119905119906119907
119904 119884
119906119907(119904) 119885
119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 119885119906119907
(119904) 119889119861 (119904)
119884119906119907
(119905 + 120575) = Γ (119883120574119905119906119907
119905+120575) 119904 isin [119905 119905 + 120575]
(54)
Using Itorsquos formula to Γ(119883120574119905119906119907
119904) we have
119889 (119884119906119907
(119904) minus Γ (119883120574119905119906119907
119904)) = 119889119884
1119906119907(119904) (55)
Combined with 119884119906119907
(119905 + 120575) minus Γ(119883120574119905119906119907
119905+120575) = 0 = 119884
1119906119907(119905 + 120575) we
get the desired result
Now consider the following BSDE
minus 1198891198842119906119907
(119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904)) 119889119904
minus 1198852119906119907
(119904) 119889119861 (119904)
(56)
1198842119906119907
(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)
Then we have the following lemma
Lemma 20 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816le 119862120575
32 119875-as (58)
where 119862 is independent of the control processes 119906 119907
Proof From Lemma 14 we know there exists some constant119862 gt 0 such that
119864[ sup119904isin[119905119879]
1003816100381610038161003816119883120574119905119906119907
(119904)10038161003816100381610038162| F119905] le 119862 (1 +
100381710038171003817100381712057411990510038171003817100381710038172) (59)
combined with
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]
le 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
119887 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119903
10038161003816100381610038161003816100381610038161003816
2
| F119905]
+ 2119864[ sup119904isin[119905119905+120575]
10038161003816100381610038161003816100381610038161003816int
119904
119905
120590 (119883120574119905119906119907
119903 119906 (119903) 119907 (119903)) 119889119861 (119903)
10038161003816100381610038161003816100381610038161003816
2
| F119905]
(60)
we have
119864[ sup119904isin[119905119905+120575]
1003816100381610038161003816119883120574119905119906119907
(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)
From (52) and (54) using Lemma 8 set
1205851 = 1205852 = 0
119892 (119904 119910 119911) = 119865 (119883120574119905119906119907
119904 119910 119911 119906119904 119907119904)
1205931 (119904) = 0
1205932 (119904) = 119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906119904 119907119904)
minus 119865 (119883120574119905119906119907
119904 119884
2119906119907(119904) 119885
2119906119907(119904) 119906119904 119907119904)
(62)
Denote 1205880(119903) = (1 + |119909|2)119903 due to 119887 120590 119891 are Lipishtiz and
that they are of linear growth Γ isin C12
119887 |1205932(119904)| le 1205880(|119883
120574119905119906119907
119904minus
120574119905(119905)|) we have
119864[int
119905+120575
119905
(100381610038161003816100381610038161198841119906119907
(119904) minus1198842119906119907
(119904)10038161003816100381610038161003816
2
+100381610038161003816100381610038161198851119906119907
(119904) minus1198852119906119907
(119904)10038161003816100381610038161003816
2
) 119889119904 | F119905]
le 119864[int
119905+120575
119905
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) 119889119904 | F119905]
le 119862120575119864[ sup119904isin[119905119905+120575]
1205882
0(1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905 (119905)
1003816100381610038161003816) | F119905]
le 1198621205752
(63)
Mathematical Problems in Engineering 9
So100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816
=10038161003816100381610038161003816119864 [(119884
1119906119907(119904) minus 119884
2119906119907(119904)) | F119905]
10038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
119864 [int
119905+120575
119905
(119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904))
minus119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]
100381610038161003816100381610038161003816100381610038161003816
le 119862119864[int
119905+120575
119905
[1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
+100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816] 119889119904 | F119905]
le 119862119864[int
119905+120575
119905
1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) 119889119904 | F119905]
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
le 11986212057532
(64)
Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation
minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]
1198840 (119905 + 120575) = 0
(65)
where 1198650(120574119905 119910 119911) = sup119906isin119880
inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as
1198840 (119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) (66)
Proof First we define a function as follows
1198651 (120574119905 119910 119911 119906) = inf119907isin119881
119865 (120574119905 119910 119911 119906 119907)
(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880
(67)
Consider the following equation
minus1198891198843119906
(119904) = 1198651 (120574119905 1198843119906
(119904) 1198853119906
(119904) 119906 (119904)) 119889119904
minus1198853119906
(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]
1198843119906
(119905 + 120575) = 0
(68)
according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906
1198853119906
) solving (68) Also
1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as for every 119906 isin U119905119905+120575
(69)
In fact according to the definition of 1198651 and Lemma 7 wehave
1198843119906
(119905) le 1198842119906119907
(119905) 119875-as for any 119907 isin V119905119905+120575
for every 119906 isin U119905119905+120575
(70)
On the other side we have the existence of a measurablefunction 119907
4 R timesR119889
times 119880 rarr 119881 such that
1198651 (120574119905 119910 119911 119906)
= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906
(71)
Set 1199074(119904) = 1199074(119884
3119906(119904) 119885
3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know
1199074isin V119905119905+120575 Therefore (1198843119906
1198853119906
) = (11988421199061199074
11988521199061199074
) which isdue to the uniqueness of the solution of (68) In particular1198843119906
(119905) = 11988421199061199074
(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-119886119904 for every 119906 isin U119905119905+120575
Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880
1198651(120574119905 119910 119911 119906) wealso derive
1198840 (119905) = esssup119906isinU119905119905+120575
1198843119906
(119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as
(72)
Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575
32 119875-as
(73)
where the constant 119862 is independent of the control processes119906 119907
Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as
100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
2
le 119862120575
119864 [int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904] le 119862120575
(74)
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
So100381610038161003816100381610038161198841119906119907
(119905) minus 1198842119906119907
(119905)10038161003816100381610038161003816
=10038161003816100381610038161003816119864 [(119884
1119906119907(119904) minus 119884
2119906119907(119904)) | F119905]
10038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816
119864 [int
119905+120575
119905
(119865 (119883120574119905119906119907
119904 119884
1119906119907(119904) 119885
1119906119907(119904) 119906 (119904) 119907 (119904))
minus119865 (120574119905 1198842119906119907
(119904) 1198852119906119907
(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]
100381610038161003816100381610038161003816100381610038161003816
le 119862119864[int
119905+120575
119905
[1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
+100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816] 119889119904 | F119905]
le 119862119864[int
119905+120575
119905
1205880 (1003816100381610038161003816119883
120574119905119906119907
119904minus 120574119905
1003816100381610038161003816) 119889119904 | F119905]
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198841119906119907
(119904) minus 1198842119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
+ 11986212057512
119864[int
119905+120575
119905
100381610038161003816100381610038161198851119906119907
(119904) minus 1198852119906119907
(119904)10038161003816100381610038161003816
2
119889119904 | F119905]
12
le 11986212057532
(64)
Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation
minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]
1198840 (119905 + 120575) = 0
(65)
where 1198650(120574119905 119910 119911) = sup119906isin119880
inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as
1198840 (119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) (66)
Proof First we define a function as follows
1198651 (120574119905 119910 119911 119906) = inf119907isin119881
119865 (120574119905 119910 119911 119906 119907)
(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880
(67)
Consider the following equation
minus1198891198843119906
(119904) = 1198651 (120574119905 1198843119906
(119904) 1198853119906
(119904) 119906 (119904)) 119889119904
minus1198853119906
(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]
1198843119906
(119905 + 120575) = 0
(68)
according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906
1198853119906
) solving (68) Also
1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as for every 119906 isin U119905119905+120575
(69)
In fact according to the definition of 1198651 and Lemma 7 wehave
1198843119906
(119905) le 1198842119906119907
(119905) 119875-as for any 119907 isin V119905119905+120575
for every 119906 isin U119905119905+120575
(70)
On the other side we have the existence of a measurablefunction 119907
4 R timesR119889
times 119880 rarr 119881 such that
1198651 (120574119905 119910 119911 119906)
= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906
(71)
Set 1199074(119904) = 1199074(119884
3119906(119904) 119885
3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know
1199074isin V119905119905+120575 Therefore (1198843119906
1198853119906
) = (11988421199061199074
11988521199061199074
) which isdue to the uniqueness of the solution of (68) In particular1198843119906
(119905) = 11988421199061199074
(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906
(119905) = essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-119886119904 for every 119906 isin U119905119905+120575
Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880
1198651(120574119905 119910 119911 119906) wealso derive
1198840 (119905) = esssup119906isinU119905119905+120575
1198843119906
(119905) = esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905) 119875-as
(72)
Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575
32 119875-as
(73)
where the constant 119862 is independent of the control processes119906 119907
Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as
100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
2
le 119862120575
119864 [int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904] le 119862120575
(74)
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Moreover from (54) we have100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816
le 119864 [int
119905+120575
119904
119865 (120574119905 1198842119906119907
(119903) 1198852119906119907
(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]
le 119862119864[int
119905+120575
119904
(1 +1003817100381710038171003817120574119905
10038171003817100381710038172+100381610038161003816100381610038161198842119906119907
(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816) 119889119903 | F119904]
le 119862120575 + 11986212057512
119864[int
119905+120575
119904
100381610038161003816100381610038161198852119906119907
(119903)10038161003816100381610038161003816
2
119889119903 | F119904]
le 119862120575 119904 isin [119905 119905 + 120575]
(75)
and applying Itorsquos formula we get 119864[int119905+120575
119905|119885
2119906119907(119904)|
2
119889119904 |
F119905] le 1198621205752 119875-as Therefore
119864[int
119905+120575
119905
(100381610038161003816100381610038161198842119906119907
(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816) 119889119904 | F119905]
le 1198621205752+ 119862120575
12(119864[int
119905+120575
119905
100381610038161003816100381610038161198852119906119907
(119904)10038161003816100381610038161003816
2
119889119903 | F119905])
12
le 11986212057532
119875-as
(76)
Now we continue the proof of Theorem 18
Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃
0le119904le120575Λ 119905+119904
According to the DPP (Theorem 16) we have
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(77)
From 119882 ge Γ on ⋃0le119904le120575
Λ 119905+119904 as well as the monotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
120574119905119906120573(119906)
119905+120575)] minus Γ (120574119905) le 0 (78)
By Lemma 19 we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) le 0 119875-as (79)
Thus from Lemma 20
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as (80)
Therefore from essinf119907isinV119905119905+120575
1198842119906119907
(119905) le 1198842119906120573(119906)
(119905) 120573 isin
B119905119905+120575 we have
esssup119906isinU119905119905+120575
essinf119907isinV119905119905+120575
1198842119906119907
(119905)
le essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198842119906120573(119906)
(119905) le 11986212057532
119875-as(81)
Thus by Lemma 21 1198840(119905) le 11986212057532
119875-as where 1198840(sdot) solvesthe ODE (65) Consequently
11986212057512
ge1
1205751198840 (119905) =
1
120575int
119905+120575
119905
1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)
Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)
(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and
Γ ge 119882 on⋃0le119904le120575
Λ 119905+119904We need to prove
sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)
Suppose it does not hold true So we have some 120579 gt 0 suchthat
1198650 (120574119905 0 0) = sup119906isinU
inf119907isin119881
119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)
and there exists a measurable function 120595 119880 rarr 119881 such that
119865 (120574119905 0 0 119906 120595 (119906)) le minus3
4120579 forall119906 isin 119880 (86)
Moreover from the DPP (Theorem 16)
Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)]
(87)
Again from119882 le Γ on⋃0le119904le120575
Λ 119905+119904 as well as themonotonicityproperty of 119866120574
119905119906120573(119906)
119905119905+120575[sdot] (Lemma 7) we get
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[Γ (119883
119906120573(119906)
119905+120575)] minus Γ (120574119905) ge 0 119875-as
(88)
Then similar to (1) from the definition of backward semi-group we have
essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
1198841119906120573(119906)
(119905) ge 0 119875-as (89)
in particular
esssup119906isinU119905119905+120575
1198841119906120595(119906)
(119905) ge 0 119875-as (90)
Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884
1119906120576120595(119906120576)(119905) ge minus120576120575
From Lemma 20 we have
1198842119906120576120595(119906120576)(119905) ge minus119862120575
32minus 120576120575 119875-as (91)
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Moreover from (54) we have
1198842119906120576120595(119906120576)(119905)
= 119864 [int
119905+120575
119905
119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885
2119906120576120595(119906120576)(119904)
119906120576(119904) 120595 (119906
120576(119904))) 119889119904 | F119905]
le 119864[int
119905+120575
119905
(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816+ 119862
1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)
100381610038161003816100381610038161003816
+119865 (120574119905 0 0 119906120576(119904) 120595 (119906
120576(119904)))) 119889119904 | F119905]
le 11986212057532
minus3
4120579120575 119875-as
(92)
From (91) and (92) we have minus11986212057512
minus 120576 le 11986212057512
minus
(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently
sup119906isin119880
inf119907isin119881
119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)
According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)
The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)
Appendix
Proof of Theorem 16 (DPP)
Proof For convenience we set
119882120575 (120574119905) = essinf120573isinB119905119905+120575
esssup119906isinU119905119905+120575
119866120574119905119906120573(119906)
119905119905+120575[119882(119883
120574119905119906120573(119906)
119905+120575)] (A1)
We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas
Lemma A1 119882120575(120574119905) is deterministic
The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here
Lemma A2 119882120575(120574119905) le 119882(120574119905)
Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows
1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)
where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the
choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)
119882120575 (120574119905) le esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)] 119875-as
(A3)
We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907
119905119905+120575[119882(119883
120574119905119906119907
119905+120575)] Then there exists a
sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that
119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575
119868120575 (120574119905 1199061 1205731 (1199061))
= sup119894ge1
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) 119875-as
(A4)
For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) +
120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894
(⋃119894minus1
119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus
119906120576
1= sum
119894ge11Γ119894
1199061
119894isin U119905119905+120575 Also from the nonanticipativity
of 1205731 we have 1205731(119906120576
1) = sum
119894ge11Γ119894
1205731(1199061
119894) and due to the
uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906
120576
1 1205731(119906
120576
1)) = sum
119894ge11Γ119894
119868120575(120574119905 1199061
119894 1205731(119906
1
119894)) 119875-as So
119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum
119894ge1
1Γ119894
119868120575 (120574119905 1199061
119894 1205731 (119906
1
119894)) + 120576
= 119868120575 (120574119905 119906120576
1 1205731 (119906
120576
1)) + 120576
= 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119882(119883
120574119905119906120576
11205731(119906120576
1)
119905+120575)] + 120576
(A5)
On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906
120576
1oplus
1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899
119882(120574119905) le esssup1199062isinU119905+120575119879
119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)
Recalling that there exists a constant 119862 isin R such that
(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862
1003817100381710038171003817120574119905 minus 120574119905
1003817100381710038171003817 for any 120574119905 120574119905 isin Λ
(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))
1003816100381610038161003816
le 1198621003817100381710038171003817120574119905 minus 120574
119905
1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879
(A7)
We can show by approximating119883120574119905119906120576
11205731(119906120576
1)
119905+120575that
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062)) 119875-as
(A8)
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
Now we estimate the right side of the latter inequality thereexists some sequence 1199062
119895 119895 ge 1 sub U119905+120575119879 such that
esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
= sup119895ge1
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A9)
Then setting Δ 119895 = esssup1199062isinU119905+120575119879
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732(1199062)) le
119869(119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732(119906
2
119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets
Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1
119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an
(ΩF119905+120575)-partition and 119906120576
2= sum
119895ge11Δ119895
1199062
119895isin U119905+120575119879 Due to
the nonanticipativity of 1205732 we get 1205732(119906120576
2) = sum
119895ge11Δ119895
1205732(1199062
119895)
also from the definitions of 1205731 1205732 we have 120573(119906120576
1oplus 119906
120576
2) =
1205731(119906120576
1) oplus 1205732(119906
120576
2) Therefore due to the uniqueness of the
solution of functional FBSDE we get
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
2 1205732 (119906
120576
2))
= 119884119883120574119905119906120576
11205731(119906120576
1)
119905+120575119906120576
21205732(119906120576
2)(119905 + 120575)
= sum
119895ge1
1Δ119895
119884119883120574119905119906120576
11205731(119906120576
1)
119905+1205751199062
1198951205732(1199062
119895)(119905 + 120575)
= sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
2
119895 1205732 (119906
2
119895)) 119875-as
(A10)
So
119882(119883120574119905119906120576
11205731(119906120576
1)
119905+120575)
le esssup1199062isinU119905+120575119879
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 1199062 1205732 (1199062))
le sum
119895ge1
1Δ119895
119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
2
119895 120573 (119906
120576
1oplus 119906
2
119895)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576
1oplus 119906
120576
2 120573 (119906
120576
1oplus 119906
120576
2)) + 120576
= 119869 (119883120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576 119875-as
(A11)
where 119906120576
= 119906120576
1oplus 119906
120576
2isin U119905119879 From (A5) (A11) and the
comparison theorem for BSDEs we know
119882120575 (120574119905)
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576)) + 120576] + 120576
le 119866120574119905119906120576
11205731(119906120576
1)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119866120574119905119906120576120573(119906120576)
119905119905+120575[119869 (119883
120574119905119906120576
11205731(119906120576
1)
119905+120575 119906
120576 120573 (119906
120576))] + (119862 + 1) 120576
= 119869 (120574119905 119906120576 120573 (119906
120576)) + (119862 + 1) 120576
= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576
le esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576 119875-as
(A12)
Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus
119882120575 (120574119905) le essinf120573isinB119905119879
esssup119906isinU119905119879
119884120574119905119906120573(119906)
(119905) + (119862 + 1) 120576
= 119882(120574119905) + (119862 + 1) 120576
(A13)
Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)
Lemma A3 119882(120574119905) le 119882120575(120574119905)
Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know
119882120575 (120574119905) = essinf1205731isinB119905119905+120575
esssup1199061isinU119905119905+120575
11986612057411990511990611205731(1199061)
119905119905+120575[119882(119883
12057411990511990611205731(1199061)
119905+120575)]
= essinf1205731isinB119905119905+120575
119868120575 (120574119905 1205731)
(A14)
and the existence of some sequence 1205731
119894 119894 ge 1 sub B119905119905+120575 such
that
119882120575 (120574119905) = inf119894ge1
119868120575 (120574119905 1205731
119894) 119875-as (A15)
For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731
119894) minus 120576 le 119882120575(120574119905) isin
F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1
119897=1Π119897) isin F119905 119894 ge
2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576
1=
sum119894ge1
1Λ119894
1205731
119894isin B119905119905+120575 combining the uniqueness of the solu-
tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576
1(1199061)) =
sum119894ge1
1Λ119894
119868120575(120574119905 1199061 1205731
119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575
119882120575 (120574119905) ge sum
119894ge1
1Π119894
119868120575 (120574119905 1205731
119894) minus 120576
ge sum
119894ge1
1Π119894
119868120575 (120574119905 1199061 1205731
119894(1199061)) minus 120576
= 119868120575 (120574119905 1199061 120573120576
1(1199061)) minus 120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
119875-as forall1199061 isin U119905119905+120575
(A16)
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899
) there exists 120573119909120576isin
B119905+120575119879 such that
119882(119909119905+120575) ge esssup1199062isinU119905+120575119879
119869 (119909119905+120575 1199062 120573119909120576
(1199062)) minus 120576 119875-as
(A17)
Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and
diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883
1205741199051199061120573120576
1(1199061)
119905+120575] = sum
119894ge1120574119894
119905+12057511198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894
we have
100381610038161003816100381610038161003816119883
1205741199051199061120573120576
1(1199061)
119905+120575minus [119883
1205741199051199061120573120576
1(1199061)
119905+120575]100381610038161003816100381610038161003816le 120576
everywhere on Ω forall1199061 isin U119905119905+120575
(A18)
And for every 120574119894 there exists some 120573
120574119894120576isin B119905+120575119879 satisfying
(A17) and 120573120576
1199061
= sum119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894120573120574119894120576isin B119905+120575119879
Next we define the new 120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) 119906 isin
U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576
U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact
let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be
such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906
1015840 into 1199061 1199061015840
1isin
U119905119905+120575 1199062 1199061015840
2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906
1015840
1and 119906 = 1199062 oplus 119906
1015840
2
From 1199061 equiv 1199061015840
1on [[119905 119878 and (119905 + 120575)]] we get 120573120576
1(1199061) = 120573
120576
1(119906
1015840
1) on
[[119905 119878and(119905+120575)]] (note that 120573120576
1is nonanticipating) On the other
side from 1199062 equiv 1199061015840
2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt
119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576
1(1199061)
119905+120575= 119883
1205741199051199061015840
1120573120576
1(1199061015840
1)
119905+120575
Thus from the definition 120573120576
1199061
= 120573120576
11990610158401
on 119878 gt 119905 + 120575 and120573120576
1199061
(1199062) = 120573120576
11990610158401
(1199061015840
2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces
120573120576(119906) = 120573
120576
1(1199061) oplus 120573
120576
1199061
(1199062) = 120573120576
1(119906
1015840
1) oplus 120573
120576
11990610158401
(1199061015840
2) on [[119905 119878]] from
which we know 120573120576isin B119905119879
For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =
119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get
119882120575 (120574119905) ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882(119883
1205741199051199061120573120576
1(1199061)
119905+120575)] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575]) minus 119862120576] minus 120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119882([119883
1205741199051199061120573120576
1(1199061)
119905+120575])] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119882(120574
119894
119905+120575)] minus 119862120576
119875-as
(A19)
Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have
119882120575 (120574119905)
ge1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))minus 120576]minus119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[sum
119894ge1
11198831205741199051199061120573120576
1(1199061)
119905+120575isin119874119894119869 (120574
119894
119905+120575 1199062 120573
120574119894120576(1199062))] minus 119862120576
= 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 ([119883
1205741199051199061120573120576
1(1199061)
119905+120575] 1199062 120573
120576
1199061
(1199062))] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062)) minus 119862120576] minus 119862120576
ge 1198661205741199051199061120573120576
1(1199061)
119905119905+120575[119869 (119883
1205741199051199061120573120576
1(1199061)
119905+120575 1199062 120573
120576
1199061
(1199062))] minus 119862120576
= 119866120574119905119906120573120576(119906)
119905119905+120575[119884
120574119905119906120573120576(119906)
(119905 + 120575)] minus 119862120576
= 119884120574119905119906120573120576(119906)
(119905) minus 119862120576 119875-as for any 119906 isin U119905119879
(A20)
Therefore
119882120575 (120574119905) ge esssup119906isinU119905119879
119869 (120574119905 119906 120573120576(119906)) minus 119862120576
ge essinf120573isinB119905119879
esssup119906isinU119905119879
119869 (120574119905 119906 120573 (119906)) minus 119862120576
= 119882(120574119905) minus 119862120576 119875-as
(A21)
Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)
Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that
119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)
119905119905+120575[119882(119883
120574119905119906120576120573(119906120576)
119905+120575)] + 120576
119875-as(A22)
(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin
119880119905119905+120575
119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)
119905119905+120575[119882(119883
120574119905119906120573120576(119906)
119905+120575)] minus 120576
119875-as(A23)
(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =
inf120573isinB119905119879
sup119906isinU119905119879
119864[119869(120574119905 119906 120573(119906))]
Acknowledgments
This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China
References
[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990
[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990
[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995
[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997
[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991
[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992
[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992
[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997
[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008
[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006
[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008
[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999
[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543
[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009
[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010
[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010
[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013
[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317
[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971
[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144
[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of