Working Paper Series

_______________________________________________________________________________________________________________________

National Centre of Competence in Research Financial Valuation and Risk Management

Working Paper No. 827

Maximum Principle for Infinite Dimensional Stochastic

Control Systems

Kai Du Qingxin Meng

First version: October 2012 Current version: October 2012

This research has been carried out within the NCCR FINRISK project on

“Mathematical Methods in Financial Risk Management”

___________________________________________________________________________________________________________

Maximum principle for infinite dimensional

stochastic control systems

Kai Du§ Qingxin Meng†

October 31, 2012

Abstract

The general stochastic maximum principle in finite dimensions is a well-known result obtained by Peng [SICON 28 (1990)]. The present paperextends this result to infinite dimensional controlled stochastic evolutionsystems. The control is allowed to take values in a nonconvex set and enterinto both drift and diÆusion terms.

Keywords. Stochastic maximum principle, infinite dimensional controlsystem, operator-valued backward stochastic diÆerential equation, general-ized solution.AMS 2010 Subject Classifications. 93E20, 49K27, 60H15.

1 Introduction

1.1 Problem formulation and basic assumptions

In this paper we shall always indicate by H a real separable Hilbert space and byh·, ·iH and k · kH its inner scalar product and norm, respectively. Denote by B(H)the Banach space of all bounded linear operators from H to itself endowed withthe norm kTkB(H) := sup{kTxkH : kxkH = 1}.

§Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland (kai.du@math.ethz.ch).Financial support by the National Centre of Competence in Research “Financial Valuation andRisk Management” (NCCR FINRISK), Project D1 (Mathematical Methods in Financial RiskManagement) is gratefully acknowledged. The NCCR FINRISK is a research instrument of theSwiss National Science Foundation.

†Department of Mathematical Sciences, Huzhou University, Zhejiang 313000, China(mqx@hutc.zj.cn). Financial support by the National Natural Science Foundation of China(11101140) and the Natural Science Foundation of Zhejiang Province (Y6110775, Y6110789) isgratefully acknowledged.

1

Let (≠,F , F, P) be a probability space with the filtration F = (Ft)t∏0 generatedby countable independent standard Wiener processes {W i; i 2 N} and augmentedwith all P-null sets of F . For simplicity, we write f ·dW =

P

i2N fi dW i with a se-quence f = (fi; i 2 N). We denote by Et[ · ] = E[ · |Ft] the conditional expectationwith respect to Ft, and by P the predictable æ-field on ≠£ [0, 1].

In this paper, we study an infinite-dimensional optimal control problem gov-erned by the following abstract semilinear stochastic evolution equation (SEE)

dx(t) = [Ax(t) + f(t, x(t), u(t))] dt + g(t, x(t), u(t)) · dWt,

x(0) = x0, (1.1)

where x(·) is the state process and u(·) is the control. The control set U is anonempty Borel-measurable subset of a metric space whose metric is denoted bydist(·, ·). Fix an element (denoted by 0) in U , and then define |u|U = dist(u, 0).An admissible control u(·) is a U -valued predictable process such that

sup©

E |u(t)|4U : t 2 [0, 1]™

< 1.

Our optimal control problem is to find an admissible control u(·) minimizing thecost functional

J(u(·)) = EZ 1

0

l(t, x(t), u(t)) dt + Eh(x(1)). (1.2)

In the above statement, A is the infinitesimal generator of a C0-semigroup, and

f : ≠£ [0, 1]£H £ U ! H, g : ≠£ [0, 1]£H £ U ! l2(H),

l : ≠£ [0, 1]£H £ U ! R, h : ≠£H ! R,

where the Hilbert space

l2(H) :=n

z = (zi)i2N : zi 2 H and kzk2l2(H) :=

X

i2Nkzik2

H < 1o

.

Throughout this paper we make the following assumptions.

Assumption 1.1. The operator A : D(A) Ω H ! H is the infinitesimal generatorof a C0-semigroup {etA 2 B(H); t ∏ 0}. Set

MA := sup{ketAkB(H) : t 2 [0, 1]}.Assumption 1.2. The functions f , g and l are all P £B(H)£B(U)-measurable,and h is F1£B(H)-measurable; for each (t, u, !) 2 [0, 1]£U £≠, f, g, l and h areglobally twice Frechet diÆerentiable with respect to x; fx, gx, fxx, gxx, lxx and hxx

are bounded by a constant M0; f, g, lx and hx are bounded by M0(1+kxkH + |u|U);l and h is bounded by M0(1 + kxk2

H + |u|2U).

2

In view of Assumption 1.1, the precise meaning of the equation (1.1) is

x(t) = etAx0 +

Z t

0

e(t°s)A£

f(s, x(s), u(s)) ds + g(s, x(s), u(s)) · dWs

§

.

A process x(·) satisfying the above equality is usually called a mild solution toequation (1.1), cf. [5].

1.2 Developments of stochastic maximum principle and

contributions of the paper

A classical approach for optimal control problems is to derive necessary conditionssatisfied by an optimal solution. As a well-known example, Pontryagin’s maximumprinciple (cf. [17]) states that any optimal control along with the correspondingstate trajectory must solve an Hamiltonian system together with a maximum con-dition of a function called the Hamiltonian. In principle, solving an Hamiltonianshould be much easier than solving the original control problem.

The original version of Pontryagin’s maximum principle was for deterministicproblems. The basic idea is to perturb an optimal control in a particular way, andconsider the first-order Taylor expansion of the state process and cost functionalaround the optimal control. By sending the perturbation to zero, one obtains akind of variational inequality and then, by duality, the maximum principle involv-ing an adjoint process. The stochastic control case, called the stochastic maximumprinple (SMP for short), was extensively studied since 1970s (see, e.g. [1, 4, 11, 14]for finite dimensional systems, and [2, 12, 20] for infinite dimensional ones). How-ever, prior to Peng’s work [16], the results were essentially obtained under thecondition that the control domain was convex or the diÆusion did not depend onthe control. The main di±culty when facing a general controlled diÆusion, roughlyspeaking, is that the Ito integral term is not of the same order (w.r.t. time) asthe Lebesgue term, and thus the usual first-order variation method fails. In thefinite dimensional case, this di±culty was overcome by Peng [16], who studied thesecond-order Taylor expansion of the spike variation. He then obtained a generalSMP for control-dependent diÆusions and not necessarily convex control domains,which involves, in addition to the first-order adjoint process, a second-order adjointprocess. These adjoint processes obtained originally by duality are then describedby backward stochastic diÆerential equations1 (BSDE). So far his work has beenextended to various control problems of finite dimension (see, e.g., [19, 18, 21]).However, the infinite dimensional extension of Peng’s result, i.e., the general SMP

1For more aspects on BSDEs, we refer to, e.g., [8]. Normally, the solution of a BSDE consistsof a pair of adapted processes.

3

for infinite dimensional control systems, remained open for a long time. The aimof this paper is to fill this gap.

The main di±culty in the infinite dimensional problem lies in the duality anal-ysis on the second-order term of a variational inequality (say (5.1)). More specifi-cally, this term is a bilinear functional on an H-valued process space; in the finitedimensional case, it can be extended as a bounded linear functional on a prod-uct space of two Hilbert spaces of matrix-valued square integrable processes2 sothat the Riesz representation theorem works, while in the infinite dimensional casethe corresponding extension fails, and thus the usual Hilbert space method is nolonger available. In this paper, we overcome the di±culty by two steps. First,we employ real analysis to calculate directly the limit of the second-order termwhen the perturbation tends to zero, and represent it via a B(H)-valued processeP (·). The Lebesgue diÆerentiation theorem and an approximation argument playkey roles in our approach (see the proofs of Lemma 5.1 and Proposition 3.5, re-spectively). Next, we prove that the second-order adjoint process, which is an“appropriate version” of eP (·), solves a B(H)-valued BSDE in the sense of what iscalled “generalized solution” which was first proposed by Guatteri-Tessitore [10] inthe study of infinite dimensional LQ problems. It is noteworthy that of the B(H)-valued BSDE the generalized solution characterizes only the first unknown whilethe complete solvability remains open. This causes another classical approach ofusing Ito formula unavailable at the moment.

Recently there are two other independent works [15, 9] concerning such a sub-ject besides ours. In the latest version of Lu-Zhang [15], a new concept of solutionto B(H)-valued BSDEs called the “relaxed transposition solution”3 was proposed,based on which they derived a SMP for optimal controls. Their result relies on somemore or less artificial conditions4, mainly due to technical purpose. Fuhrman et al.[9] formulated a SMP for the system governed by a concrete equation (stochasticparabolic PDE with deterministic coe±cients). Their approach depends on thespecial structure of the equation. In our work, the setting is quite general, andconsistent with the finite dimensional case.

The rest of this paper is structured as follows. In Section 2, we present severalpreliminary results and then state our main theorem, the SMP. Section 3 is devotedto the investigation of a stochastic bilinear functional. In Section 4, we prove theexistence and uniqueness of the generalized solution to a B(H)-valued BSDE.Finally, Section 5 completes the proof of the SMP.

We finish the introduction with some notations. Let H be a separable realHilbert space and B(H) be its Borel æ-field. The following classes of processes will

2In this case, the matrix norm is taken as the Frobenius norm.3This definition also does not include a full characterization of the second unknown part.4See [15], the end of Section 1, and Remark 9.1, for more explanations.

4

be used in this article. Here p 2 [1,1].• Lp

P(≠£[0, 1]; H) : the space of equivalence classes of F£B([0, 1]) -measurableprocesses x(·) admitting a predictable version such that E

R 1

0 kx(t)kpHdt < 1.

• LpP(≠; C([0, 1]; H)) : the space of predictable processes x(·) with continuous

paths in H such that E supt2[0,1] kx(t)kpH < 1. Elements of this space are defined

up to indistinguishability.• Lp

F([0, 1]; Lp(≠; H)) : the space of equivalence classes of F-adapted processesx(·) such that x(·) : [0, 1] ! Lp(≠; H) is B([0, 1])-measurable and

R 1

0 Ekx(t)kpH dt <

1.• CF([0, 1]; Lp(≠; H)) : the space of H-valued F-adapted processes x(·) such

that x(·) : [0, 1] ! Lp(≠; H) is strongly continuous and supt2[0,1] Ekx(t)kpH < 1.

Elements of this space are defined up to modification.Moreover, since the æ-field generated by the operator norm in B(H) is too

large, we shall define the following spaces with respect to B(H)-valued processesand random variables.

• LpPw(≠ £ [0, 1]; B(H)) : the space of equivalence classes of B(H)-valued

processes T (·) such that hx, T (·)yiH 2 LpP(≠ £ [0, 1]; R) for any x, y 2 H, and

ER 1

0 kT (t)kpB(H)dt < 1. Here the subscript “w” stands for “weakly measurable”.

• CF([0, 1]; Lpw(≠; B(H))) : the space of B(H)-valued processes T (·) such that

hx, T (·)yiH 2 CF([0, 1]; Lp(≠; R)) for any x, y 2 H, and supt2[0,1] EkT (t)kpB(H) < 1.

Elements of this space are defined up to modification.• Lp

Ftw(≠; B(H)) : the space of equivalence classes of B(H)-valued randomvariable T such that hx, TyiH 2 Lp

Ft(≠; H) for any x, y 2 H, and EkTkp

B(H) < 1.

2 Preliminaries and the main result

2.1 Auxiliary results on SEEs and BSEEs

Let b and æ be two P£B(H)-measurable functions with values in H and l2(H),respectively, and F be a P £ B(H)£B(l2(H))-measurable function with values inH, such that

kb(t, x)° b(t, x)kH + kæ(t, x)° æ(t, x)kl2(H) ∑ M0kx° xkH ,

kF (t, x, y)° F (t, x, y)kH ∑ M0(kx° xkH + ky ° ykl2(H))

for a constant M0 > 0 and any t 2 [0, 1], x, x 2 H and y, y 2 l2(H).For given operator A satisfying Assumption 1.1, consider the following SEE

x(t) = etAx0 +

Z t

0

e(t°s)A£

b(s, x(s)) ds + æ(s, x(s)) · dWs

§

(2.1)

5

and BSEE

p(t) = e(1°t)A§ª +

Z 1

t

e(s°t)A§£

F (s, p(s), q(s)) ds° q(s) · dWs

§

. (2.2)

We present some preliminary results on SEE (2.1) and BSEE (2.2) which willbe often used in this paper (cf. [5, 13]).

Lemma 2.1. Under the above setting, we have the following assertions:(1) If b(·, 0) 2 Lp

P(≠£[0, 1]; H) and æ(·, 0) 2 LpP(≠£[0, 1]; l2(H)) for p 2 [2,1),

then SEE (2.1) has a unique solution x(·) in the space LpP(≠; C([0, 1]; H)) for any

given x0 2 H, with the Lp-estimate

E supt2[0,1]

kx(t)kpH ∑ K E

∑

kx0kpH +

≥

Z 1

0

kb(t, 0)kH dt¥p

+≥

Z 1

0

kæ(t, 0)k2l2(H) dt

¥

p2

∏

,

where the constant K depends only on MA,M0 and p.(2) If F (·, 0, 0) 2 L2

P(≠ £ [0, 1]; H), then BSEE (2.2) has a unique solution(p(·), q(·)) in the space L2

P(≠; C([0, 1]; H)) £ L2P(≠ £ [0, 1]; l2(H)) for any given

ª 2 L2F1

(≠; H).

The following lemma, concerning the duality between SEE (2.1) and BSEE(2.2), can be found in [12].

Lemma 2.2. Under the conditions in Lemma 2.1, we have

E hx(t1), p(t1)iH + EZ t2

t1

h

hb(s, x(s)), p(s)iH + hæ(s, x(s)), q(s)il2(H)

i

ds

= E hx(t2), p(t2)iH + EZ t2

t1

hx(s), F (s, p(s), q(s))iH ds

for any 0 ∑ t1 ∑ t2 ∑ 1.

2.2 Well-posedness of an operator-valued BSDE

In order to characterize the second order adjoint process of the controlled system,we introduce the following operator-valued BSDE (OBSDE)

dP (t) = ° ©

A§P (t) + P (t)A + A§] (t)P (t) + P (t)A](t)

+ Tr[C§PC + QC + C§Q](t) + G(t)™

dt + Q(t) · dWt,

P (1) = ™ (2.3)

6

with the unknown processes P (·) and Q(·), where A satisfies Assumption 1.1, A](·)and C(·) are given coe±cients, and

Tr[C§PC + QC + C§Q](t) =X

i2N[C§

i PCi + QiCi + C§i Qi](t).

We call the pair (G, ™) the input of OBSDE (2.3).

Assumption 2.3. A](·) 2 L1Pw(≠ £ [0, 1]; B(H)). C(·) = (Ci(·) ; i 2 N) withCi(·) 2 L1Pw(≠£ [0, 1]; B(H)). Assume

ess supt,!

n

kA](t,!)k2B(H),

X

i2NkCi(t,!)k2

B(H)

o

∑ M0.

The solvability theory of such a B(H)-valued BSDE is still far from complete.A first remarkable work on such a subject was found in Guatteri-Tessitore [10]where, inspired by the notion of “strong solution” for PDEs (cf. [3]), the authorsproposed for an OBSDE the concept of generalized solution which only involved thefirst unknown P (·), and obtained the corresponding existence-uniqueness result.This concept bases on the following lemma concerning the solvability of (2.3) inthe Hilbert space B2(H) of all Hilbert-Schmidt operators from H to itself (cf. [10,Theorem 5.4]).

Lemma 2.4 (mild solution). Under Assumptions 1.1 and 2.3, if ™ 2 L2F1

(≠; B2(H))and G 2 L2

P(≠£ [0, 1]; B2(H)), then there exists a unique pair

(P (·), Q(·)) 2 L2P(≠; C([0, 1]; B2(H)))£ L2

P(≠£ [0, 1]; l2(B2(H)))

such that for all t 2 [0, 1],

P (t) = e(1°t)A§™e(1°t)A +

Z 1

t

e(s°t)A§ [A§]P + PA] + G](s)e(s°t)A ds

+

Z 1

t

e(s°t)A§Tr[C§PC + QC + C§Q](s)e(s°t)A ds

°Z 1

t

e(s°t)A§Q(s)e(s°t)A · dWs, (a.s.). (2.4)

We call (P (·), Q(·)) the mild solution to OBSDE (2.3).

Following the spirit of Guatteri-Tessitore [10], we give the definition as below.

Definition 2.5 (generalized solution). P (·) 2 L2Pw(≠ £ [0, 1]; B(H)) is called a

generalized solution to OBSDE (2.3) if there exists a sequence (P n, Qn, Gn) satis-fying the following four conditions:

7

1) P n(1) 2 L2F1

(≠; B2(H)) and Gn 2 L2P(≠£[0, 1]; B2(H)), such that kP n(1)kB(H)

and kGn(·)kB(H) are uniformly dominated by some ¥ 2 L2F1

(≠; R) and a(·) 2L2P(≠£ [0, 1]; R), respectively.

2) (P n, Qn) is a mild solution to OBSDE (2.3) with the input (Gn, P n(1)).3) for any given x, y 2 H,

hx, P n(1,!)yiH ! hx, ™(!)yiH and hx,Gn(t,!)yiH ! hx,G(t,!)yiHfor each (t,!) 2 [0, 1]£ ≠, as n !1.

4) for any given x, y 2 H, hx, P n(t,!)yiH ! hx, P (t,!)yiH for a.e. (t,!).

In the above definition only the process P (·) is characterized. Nevertheless,this is su±cient and even natural for our optimal control problem since Q(·) isnot involved in the formulation of the SMP (see Theorem 2.7). Now we state thefollowing existence-uniqueness result on the generalized solution to OBSDE (2.3),whose proof will be given in Section 4.

Theorem 2.6. Under Assumptions 1.1 and 2.3, there exists a unique generalizedsolution P (·) to OBSDE (2.3) for any given ™ 2 L2

F1w(≠; B(H)) and G 2 L2Pw(≠£

[0, 1]; B(H)).

2.3 The statement of the SMP

The Hamiltonian is defined as

H(t, x, u, p, q) := l(t, x, u) + hp, f(t, x, u)iH + hq, g(t, x, u)il2(H) . (2.5)

This is a map from ≠ £ [0, 1] £ H £ U £ H £ l2(H) into R. Our main result isstated as follows.

Theorem 2.7 (stochastic maximum principle). Let Assumptions 1.1 and 1.2 besatisfied, x(·) be the state process with respect to an optimal control u(·). Then foreach u 2 U , the variational inequality

0 ∑ H(t, x(t), u, p(t), q(t))°H(t, x(t), u(t), p(t), q(t))

+1

2hg(t, x(t), u)° g(t, x(t), u(t)), P (t)[g(t, x(t), u)° g(t, x(t), u(t))]il2(H) (2.6)

holds for a.e. (t,!) 2 [0, 1) £ ≠, where (p(·), q(·)) is the solution to BSEE (2.2)with

F (t, p, q) = Hx(t, x(t), u(t), p, q) , ª = hx(x(1)),

and P (·) is the generalized solution to OBSDE (2.3) with

A](t) = fx(t, x(t), u(t)), C(t) = gx(t, x(t), u(t)),

G(t) = Hxx(t, x(t), u(t), p(t), q(t)), ™ = hxx(x(1)).(2.7)

8

The proof of this theorem is postponed to the final section. Here we give someremarks.

Remark 2.8. (1) By introducing a new function

H(t, x, u) := °H(t, x, u, p(t), q(t)° P (t)g(t, x, u(t)))

° 1

2hg(t, x, u), P (t)g(t, x, u)il2(H) ,

we can rewrite the maximum condition (2.6) as

H(t, x(t), u(t))) = maxu2U

H(t, x(t), u).

(2) Let us single out two important special cases: i) The diÆusion does notcontain the control variable, i.e., g(t, x, u) ¥ g(t, x). In this case, (2.6) becomes

H(t, x(t), u(t), p(t), q(t)) = minu2U

H(t, x(t), u, p(t), q(t)).

This is a well known result (cf. [12]). ii) The control domain (a subset of aseparable Hilbert space U) is convex and all the coe±cients are C1 in u. Then from(2.6) we can deduce

hHu(t, x(t), u(t), p(t), q(t)), u° u(t)iU ∏ 0 8u 2 U, a.e. (t,!).

This is called a local form of the maximum principle, coinciding with the result ofBensoussan [2].

3 A stochastic bilinear functional: representa-

tion and properties

Our approach begins from an investigation of a stochastic bilinear functional. LetAssumptions 1.1 and 2.3 be satisfied. It follows from Lemma 2.1 that the followingequation

yø,ª(t) = e(t°ø)Aª +

Z t

ø

e(t°s)AA](s)yø,ª(s)ds

+

Z t

ø

e(t°s)AC(s)yø,ª(s) · dWs, t 2 [ø, 1] (3.1)

has a unique solution yø,ª(·) for any given ø 2 [0, 1) and ª 2 L4Fø

(≠; H). Weindicate that Lemma 2.1 implies

Eø supt2[ø,1]

∞

∞yø,ª(t)∞

∞

4

H∑ K kªk4

H , (a.s.). (3.2)

Hereafter K is a constant depending only on MA and M0.

9

Notation 3.1. For given ™ 2 L2F1w(≠; B(H)) and G 2 L2

Pw(≠£ [0, 1]; B(H)), wewrite

§ = k™k2B(H) +

Z 1

0

kG(t)k2B(H) dt,

and (§t)t∏0 to be the continuous modification of the martingale (Et§)t∏0 with §0 =E§. Set

Yø (ª, ≥) :=≠

yø,ª(1), ™yø,≥(1)Æ

H+

Z 1

ø

≠

yø,ª(t), G(t)yø,≥(t)Æ

Hdt

for any ø 2 [0, 1) and ª, ≥ 2 L4Fø

(≠; H).

Now we define, for each ø 2 [0, 1), a stochastic bilinear functional on (L4Fø

(≠; H))2

as follows:eTø (ª, ≥) = Eø [Yø (ª, ≥)]. (3.3)

It follows from (3.2) thatØ

Ø eTø (ª, ≥)Ø

Ø ∑ Kp

§ø kªkH k≥kH 2 L1Fø

(≠; R), (3.4)

which means that eTø (·, ·) is a bilinear mapping from (L4Fø

(≠; H))2 to L1Fø

(≠; R).In particular, for any x, y 2 H,

eTø (x, y) 2 L2Fø

(≠; R).

We have the following lemmas.

Lemma 3.2. For any given ø 2 [0, 1), x, y 2 H and sets E1, E2 2 Fø , we haveeTø (x · 1E1 , y · 1E2) = 1E1\E2 · eTø (x, y).

The proof is quite direct and so omitted here.

Lemma 3.3. The process eT·(x, y) 2 CF([0, 1]; L2(≠; R)) for any x, y 2 H.

Proof. First we prove that eT·(x, y) 2 CF([0, 1]; L1(≠; R)) for any x, y 2 H. DenoteYt = Yt(x, y). We have

lims!t

E |Es[Ys]° Et[Yt]| = lims!t

E |Es[Ys ° Yt]° (Et[Yt]° Es[Yt])|∑ lim

s!tE |Ys ° Yt| + lim

s!tE |Et[Yt]° Es[Yt]| .

Without loss of generality, we assume t < s. On the one hand, the process(Er[Yt])r∏0 is a uniformly integrable martingale, thus it follows from the Doobmartingale convergence theorem (cf. [6])

lims!t

E |Et[Yt]° Es[Yt]| = 0.

10

On the other hand, note that

|Ys ° Yt| ∑Ø

Øhys,x(1), ™ys,y(1)iH °≠

yt,x(1), ™yt,x(1)Æ

H

Ø

Ø

+

Z 1

s

Ø

Øhys,x(r), G(r)ys,y(r)iH °≠

yt,x(r), G(r)yt,y(r)Æ

H

Ø

Ø dr

+

Z s

t

Ø

Ø

≠

yt,x(r), G(r)yt,y(r)Æ

H

Ø

Ø dr

=: I1 + I2 + I3.

First, it follows from (3.2) and Young’s inequality that

|EI3|2 ∑ K§0 |t° s| kxk2H kyk2

H ! 0, (3.5)

as |s° t| ! 0. Next, from (3.2) and the continuity of the solution to SEEs, wehave

|EI2|2 ∑ K§0

≥

kxk2H

q

E ky ° yt,y(s)k4H

+ kyk2H

q

E kx° yt,x(s)k4H

¥

! 0, as |s° t|! 0.

Similarly, we can show

|EI1|2 ! 0, as |s° t|! 0.

Therefore, we have for any x, y 2 H,

lims!t

EØ

Ø eTs(x, y)° eTt(x, y)Ø

Ø = 0. (3.6)

This implies eT·(x, y) 2 CF([0, 1]; L1(≠; R)).Next, it follows from (3.4) and the Doob martingale convergence theorem that

for any x, y 2 H

|eTs(x, y)° eTt(x, y)|2 ∑ Kkxk2Hkyk2

H(§s + §t)

L1°! Kkxk2Hkyk2

H · 2§t as s ! t.

This along with (3.6) and the Lebesgue dominated convergence theorem yields

lims!t

EØ

Ø eTs(x, y)° eTt(x, y)Ø

Ø

2= 0. (3.7)

The lemma is proved.

Thanks to the Riesz representation theorem, we have the following result.

11

Lemma 3.4. There is a unique eP (·) in the space CF([0, 1]; L2w(≠; B(H))) such that

hª, eP (ø)≥iH = eTø (ª, ≥) (a.s.) for any given ø 2 [0, 1) and ª, ≥ 2 L4Fø

(≠; H).

Proof. Fix arbitrary ø 2 [0, 1) and take a standard complete orthonormal basis{eH

i } in H. Then there is a set of full probability ≠ø Ω ≠ such that for each! 2 ≠ø ,

Ø

Ø

£

eTø (eHi , eH

j )§

(!)Ø

Ø ∑p

§ø (!), 8i, j 2 N.

Hence, from the Riesz representation theorem, there is a unique eP (ø,!) 2 B(H)for each ! 2 ≠ø such that

≠

eHi , eP (ø,!)eH

j

Æ

H=

£

eTø (eHi , eH

j )§

(!), 8i, j 2 N. (3.8)

It follows from the bilinearity that hx, eP (ø)yiH = eTø (x, y) (a.s.) for any x, y 2 H.Then from Lemma 3.2 and by an argument of approximation, one can show thathª, eP (ø)≥iH = eTø (ª, ≥) (a.s.) for any ª, ≥ 2 L4

Fø(≠; H), and

|hª, eP (ø)≥iH | ∑ Kp

§ø kªkH k≥kH (a.s.). (3.9)

Finally, the fact that eT·(x, y) 2 CF([0, 1]; L2(≠; R)) for any x, y 2 H implies eP (·) 2CF([0, 1]; L2

w(≠; B(H))). The uniqueness is direct. The lemma is proved.

The following property will play a key role in our proof of SMP.

Proposition 3.5. Let the conditions in Theorem 2.6 be satisfied. For given ø 2[0, 1), # 2 L4

Fø(≠; l2(H)) and " 2 (0, 1 ° ø), let yø,#

" (·) be the mild solution toequation

yø,#" (t) =

Z t

ø

e(t°s)AA](s)yø,#" (s) ds

+

Z t

ø

e(t°s)Ah

C(s)yø,#" (s) + "°

12 1[ø,ø+")#

i

· dWs, t 2 [ø, 1].

Then we have

E≠

#, eP (ø)µÆ

l2(H)= lim

"#0T "

ø (#, µ)

:= lim"#0

E∑

≠

yø,#" (1), ™yø,µ

" (1)Æ

H+

Z 1

ø

≠

yø,#" (t), G(t)yø,µ

" (t)Æ

Hdt

∏

for any ø 2 [0, 1), #, µ 2 L4Fø

(≠; l2(H)).

12

The above proposition follows from an approximation argument which we di-vide into the subsequent lemmas. First of all, we make several notations. Define

BA(ø) :=©

ª 2 L4Fø

(≠; H) : ª(!) 2 D(A)

and kªkA := sup!

(kAªkH + kªkH) < 1™

which is dense in L4Fø

(≠; H). Set ei = (0, . . . , 0, 1, 0, . . . ) with only the i-th elementnonzero. Then ªei 2 L4

Fø(≠; l2(H)) for any ª 2 L4

Fø(≠; H) and i 2 N. Moreover,

we define

ªi(t) := "°12 (W i

t °W iø )ª and ≥j(t) := "°

12 (W j

t °W jø )≥.

Lemma 3.6. For any ø 2 [0, 1), ª 2 BA(ø) and i 2 N, we have

Ekyø,ªei" (ø + ")° ªi(ø + ")k4

H ∑ K"2kªk4A.

Proof. Set yi(t) := yø,ªei" (t). Then for t 2 [ø, ø + "],

(yi ° ªi)(t) =

Z t

ø

e(t°s)A£

A](s)(yi ° ªi)(s) + (A + A](s))ª

i(s)§

ds

+

Z t

ø

e(t°s)A£

C(s)(yi ° ªi)(s) + C(s)ªi(s)§ · dWt.

By Lemma 2.1, we have

E∞

∞(yi ° ªi)(ø + ")∞

∞

4

H∑ K E

≥

Z ø+"

ø

∞

∞ªi(t)∞

∞

2

Adt

¥2∑ K"2 kªk4

A .

The lemma is proved.

Notice that for any ª, ≥ 2 BA(ø),

T "ø (ªei, ≥ej) = E

Z ø+"

ø

≠

yø,ªei" (t), G(t)yø,≥ej

" (t)Æ

Hdt

+ E≠

yø,ªei" (ø + "), eP (ø + ")yø,≥ej

" (ø + ")Æ

H

=: J1 + J2.

Now we let " tend to 0. On the one hand, one can show that the term J1 tendsto 0 similarly as in (3.5); on the other hand, by means of Lemma 3.6, the term J2

should tend to the same limit as Ehªi(ø + "), eP (ø + ")≥j(ø + ")iH . Indeed, we have

Lemma 3.7. For any ø 2 [0, 1), ª, ≥ 2 BA(ø) and i, j 2 N, we have

lim"#0

n

E≠

ªi(ø + "), eP (ø + ")≥j(ø + ")Æ

H° T "

ø (ªei, ≥ej)o

= 0.

13

Proof. It is su±cient to show

lim"#0

n

E≠

ªi(ø + "), eP (ø + ")≥j(ø + ")Æ

H° J2

o

= 0.

Indeed, from (3.9), Lemmas 2.1 and 3.6, we have

Ø

ØE≠

ªi(ø + "), eP (ø + ")≥j(ø + ")Æ

H° J2

Ø

Ø

∑ K°

Ekªk4H

¢

14°

Ekyø,≥ej" (ø + ")° ≥j(ø + ")k4

H

¢

14

+ K°

Ek≥k4H

¢

14°

Ekyø,ªei" (ø + ")° ªi(ø + ")k4

H

¢

14

! 0, as " # 0.

This concludes this lemma.

On the other hand, from the continuity of eP (·) we have the following

Lemma 3.8. For any ø 2 [0, 1), ª, ≥ 2 BA(ø) and i, j 2 N, we have

lim"#0

E≠

ªi(ø + "), eP (ø + ")≥j(ø + ")Æ

H= E

≠

ª, eP (ø)≥Æ

H.

Proof. It is easy to see

E≠

ªi(ø + "), eP (ø)≥j(ø + ")Æ

H= E

≠

ª, eP (ø)≥Æ

H.

Thus we need prove

lim"#0

E≠

ªi(ø + "), [ eP (ø + ")° eP (ø)]≥j(ø + ")Æ

H= 0.

Indeed, we have

Ø

Ø E≠

ªi(ø + "), [ eP (ø + ")° eP (ø)]≥j(ø + ")Æ

H

Ø

Ø

2

=Ø

Ø E£

"°1(W iø+" °W i

ø )(Wjø+" °W j

ø )≠

ª, [ eP (ø + ")° eP (ø)]≥Æ

H

§

Ø

Ø

2

∑ "°2 · E°|W iø+" °W i

ø |2|W jø+" °W j

ø |2¢ · EØ

Ø

≠

ª, [ eP (ø + ")° eP (ø)]≥Æ

H

Ø

Ø

2

∑ 3EØ

Ø

≠

ª, [ eP (ø + ")° eP (ø)]≥Æ

H

Ø

Ø

2.

From the weak continuity of eP (·), one can show (as prove (3.7)) that E|hª, [ eP (ø +")° eP (ø)]≥iH |2 tends to 0 as " # 0. This concludes the lemma.

Now we are in a position to complete the proof of Proposition 3.5.

14

Proof of Proposition 3.5. Fix any #, µ 2 L4Fø

(≠; l2(H)). For arbitrary ± > 0,we can find (from the density) a large m± and {ªi, ≥i ; i = 1, . . . , m±} Ω BA(ø) suchthat

#m±:= (ª1, . . . , ªm±

), µm±:= (≥1, . . . , ≥m±

),

E k#° #m±k4

l2(H) + E kµ ° #m±k4

l2(H) < ±4.

Then we have

Ø

ØEh#, eP (ø)µil2(H) ° Eh#m±, eP (ø)µm±

il2(H)

Ø

Ø < K(#, µ, P ) ±,

|T "ø (#, µ)° T "

ø (#m±, µm±

)| < K(#, µ, P ) ±.

On the other hand, from Lemmas 3.7 and 3.8, one can easily check that

Eh#m±, eP (ø)µm±

il2(H) = lim"#0

T "ø (#m±

, µm±).

Thus we have

lim sup"#0

Ø

ØEh#, eP (ø)µil2(H) ° T "ø (#, µ)

Ø

Ø < K(#, µ, P ) ±.

From the arbitrariness of ±, we conclude the proposition. §

4 Proof of Theorem 2.6

In this section we will show that an “appropriate version” of eP (·) (determined inLemma 3.4) gives the (unique) generalized solution to OBSDE (2.3). First we have

Lemma 4.1. There is a unique P (·) in the space L2Pw(≠£ [0, 1]; B(H)) such that

hx, P (·)yiH is equivalent to eT·(x, y) and hx, eP (·)yiH in the space L2F([0, 1]; L2(≠; R))

for any x, y 2 H.

The proof bases on the following result.

Lemma 4.2. For any X(·) 2 CF([0, 1]; L2(≠; R)), there is a predictable processeX(·) which is equivalent to X(·) in the space L2

F([0, 1]; L2(≠; R)).

Proof. It is clear that X(·) 2 L2F([0, 1]; L2(≠; R)). Define

Xn(t) :=X

1∑k∑n

X≥k ° 1

n

¥

· 1( k°1n , k

n ](t) for n 2 N,

15

which is a predictable process. It follows from the continuity that

limn!1

Z 1

0

E |Xn(t)°X(t)|2 dt = 0.

This implies that {Xn(·)}n2N is a Cauchy sequence in L2P(≠£ [0, 1]; R), and then

converges to an eX(·) 2 L2P(≠£ [0, 1]; R) Ω L2

F([0, 1]; L2(≠; R)). From the unique-ness of the limit, X(·) and eX(·) are equivalent in L2

F([0, 1]; L2(≠; R)).

Proof of Lemma 4.1. Take a standard complete orthonormal basis {eHi } in H. By

means of the above lemma, we can find for each i, j 2 N a process T·(eHi , eH

j ) 2L2P(≠£ [0, 1]; R) which is equivalent to eT·(eH

i , eHj ) in L2

F([0, 1]; L2(≠; R)). Set

°ij = {(t,!) 2 [0, 1]£ ≠ :£

Tt(eHi , eH

j )§

(!) ∑p

§t(!)}.

Then °ij is a predictable set with full measure, and so ° := \i,j2N °ij is also apredictable set with full measure. Thanks to the Riesz representation theorem,there is a unique P (t,!) 2 B(H) for each (t,!) 2 ° such that

≠

eHi , P (t,!)eH

i

Æ

H=

£

Tt(eHi , eH

j )§

(!)

and kP (t,!)kB(H) ∑p

§t(!). Let P (t,!) = 0 for (t,!) 2 °c. Then heHi , P (·)eH

j iHbelongs to L2

P(≠£ [0, 1]; R). Notice that for each (t,!) and x, y 2 H,

hx, P (t,!)yiH = limn!1

hxn, P (t,!)yniH

with xn =Pn

i=1hx, eiiHei and yn =Pn

i=1hy, eiiHei. This implies hx, P (·)yiH 2L2P(≠£ [0, 1]; R). The uniqueness is obvious. The lemma is proved.

Then by an argument of approximation, we have

Corollary 4.3. For any ª(·) 2 L4P(≠ £ [0, 1]; H), hª(·), P (·)ª(·)iH belongs to

L1P(≠£[0, 1]; R), and is equivalent to hª(·), eP (·)ª(·)iH in the space L1

F([0, 1]; L1(≠; R)).

Now we are in a position to prove Theorem 2.6.

Proof of Theorem 2.6. Existence. We claim that the predictable representationP (·) of the stochastic bilinear functional eT·, which is constructed in Lemma 4.1,is the desired generalized solution to OBSDE (2.3). Indeed, for each n 2 N, weintroduce the finite dimensional projection ¶n : H ! H : x ! Pn

i=1hx, eHi iHeH

i ,define

Gn(t,!) := ¶nG(t,!)¶n and ™n(!) := ¶n™(!)¶n

16

and find from Lemma 2.4 the (unique) mild solution (P n, Qn) to OBSDE (2.3)with the input (Gn, ™n). Obviously, the conditions (1), (2) and (3) in Definition2.5 are satisfied. It remains to verify the condition (4). Let us recall (3.3), theconstruction of eTt(·, ·) for each t 2 [0, 1]. Herein we write eTt(·, ·) as eTt(·, ·; G, ™)to underline its dependence on the input (G, ™) of OBSDE (2.3), and analogouslydefine eTt(·, ·; Gn, ™n). Then by the Ito formula and the Lebesgue dominated con-vergence theorem, we can obtain for any t 2 [0, 1], x, y 2 H,

hx, P n(t)yiH = eTt(x, y; Gn, ™n)

L1°! eTt(x, y; G, ™) = hx, eP (t)yiH (4.1)

with eP (·) determined by Lemma 3.4. This along with Lemma 4.1 implies for anyx, y 2 H,Z 1

0

EØ

Øhx, P n(t)yiH ° hx, P (t)yiHØ

Ødt =

Z 1

0

EØ

Øhx, P n(t)yiH ° hx, eP (t)yiHØ

Ødt ! 0,

thus, there is a subsequence (still denoted by P n) such that hx, P n(t,!)yiH !hx, P (t,!)yiH for a.e. (t,!). The existence is proved.

Uniqueness. For each approximation sequence (P n, Qn, Gn), one can show, asin (4.1), that

hx, P n(t)yiH L1°! eTt(x, y; G, ™), 8 t 2 [0, 1];

on the other hand, the conditions (1, 4) in Definition 2.5 yields hx, P n(·)yiH con-verges to hx, P (·)yiH in L1(≠£ [0, 1]). These mean that every generalized solutionP (·) satisfies that

Z 1

0

EØ

Ø eTt(x, y; G, ™)° hx, P (t)yiHØ

Ødt = 0 8x, y 2 H,

which yields the uniqueness. The proof of Theorem 2.6 is complete.

5 Proof of Theorem 2.7

The proof is divided into the following three steps.Step 1. The spike variation and second-order expansion.The approach in this step is quite standard (cf. [16]). Recall that x(·) is the

state process with respect to a optimal control u(·). For any given admissiblecontrol u(·), we construct a perturbed admissible control in the following way

uø,"(t) :=

Ω

u(t), if t 2 [ø, ø + "],u(t), otherwise,

17

with fixed ø 2 [0, 1), su±ciently small positive ".Let xø,"(·) be the state process with respect to control uø,"(·). For the sake of

convenience, we denote for ' = f, g, l, fx, gx, lx, fxx, gxx, lxx,

'(t) := '(t, x(t), u(t)),

'¢(t) := '(t, x(t), u(t))° '(t, x(t), u(t)),

'¢,ø,"(t) := '¢(t) · 1[ø,ø+"](t),

Let x1(·) and x2(·) be the solutions respectively to

x1(t) =

Z t

0

e(t°s)Afx(s)x1(s) ds +

Z t

0

e(t°s)A[gx(s)x1(s) + g¢,ø,"(s)] · dWs,

x2(t) =

Z t

0

e(t°s)Ah

fx(s)x2(s) +1

2fxx(s) (x1 ≠ x1) (s) + f¢,ø,"(s)

i

ds

+

Z t

0

e(t°s)Ah

gx(s)x2(s) +1

2gxx(s) (x1 ≠ x1) (s) + g¢,ø,"

x (s)x1(s)i

· dWs.

It follows from Lemma 2.1 that for all t 2 [0, 1],8

>

>

>

>

<

>

>

>

>

:

"°2E kx1(t)k4H + "°1E kx1(t)k2

H + "°2E kx2(t)k2H ∑ K,

"°2E kxø,"(t)° x(t)k4H + "°1E kxø,"(t)° x(t)k2

H ∑ K,

"°2E kxø,"(t)° x(t)° x1(t)k2H ∑ K,

"°2E kxø,"(t)° x(t)° x1(t)° x2(t)k2H = o(1).

This along with the factJ(uø,"(·))° J(u(·)) ∏ 0

yields (for details, we refer to [22] or [7]) the following variational inequality

o(") ∑ EZ 1

0

h

l¢,ø,"(t) +≠

lx(t), x1(t) + x2(t)Æ

H+

1

2

≠

x1(t), lxx(t)x1(t)Æ

H

i

dt

+ E hhx(x(1)), x1(1) + x2(1)iH +1

2hx1(1), hxx(x(1))x1(1)iH . (5.1)

Step 2. First-order duality analysis.It follows from Lemma 2.1(2) that BSEE (2.2) has a unique solution (p(·), q(·))

in this situation. Recalling the Hamiltonian (2.5), and from Lemma 2.2, we have

EZ 1

0

£

l¢,ø,"(t) +≠

lx(t), x1(t) + x2(t)Æ

H

§

dt + E hhx(x(1)), x1(1) + x2(1)iH

= o(") + EZ ø+"

ø

[H(t, x(t), u(t), p(t), q(t))°H(t, x(t), u(t), p(t), q(t))] dt

+1

2E

Z 1

0

h

≠

p(t), fxx(t) (x1 ≠ x1) (t)Æ

H+ hq(t), gxx(t) (x1 ≠ x1) (t)il2(H)

i

dt.

18

By denoting

H¢(t) := H(t, x(t), u(t), p(t), q(t))°H(t, x(t), u(t), p(t), q(t)),

we get

o(1) ∑ "°1EZ ø+"

ø

H¢(t) dt +1

2"°1E hx1(1), hxx(x(1))x1(1)iH

+1

2"°1E

Z 1

0

hx1(t),Hxx(t, x(t), u(t), p(t), q(t))x1(t)iH dt. (5.2)

Step 3. Second-order duality analysis and completion of the proof.This is the key step in the proof. It follows from Theorem 2.6 that OBSDE (2.3)

with the coe±cients (2.7) has a unique generalized solution P (·). Accordingly, wecan find from Lemma 3.4 a eP (·) in this situation.

Now we fix arbitrary predictable version eg¢(·) of g¢(·). Without loss of gen-erality, we assume that Ekeg¢(ø)k4

l2(H) < 1 for each ø 2 [0, 1]. Now we introducethe following equation

yø,"(t) =

Z t

ø

e(t°s)AA](s)yø,"(s)ds

+

Z t

ø

e(t°s)A[C(s)yø,"(s) + "°12 1[ø,ø+")eg

¢(ø)] · dWs, t 2 [ø, 1].

Then we have

Lemma 5.1. For a.e. ø 2 [0, 1), we have

lim"#0

supt2[ø,1]

E∞

∞"°12 x1(t)° yø,"(t)

∞

∞

4

H= 0.

Proof. By Lemma 2.1, we have for each ø 2 [0, 1],

supt2[ø,1]

E∞

∞"°12 x1(t)° yø,"(t)

∞

∞

4

H

∑ K

"

Z 1

ø

E∞

∞

eg¢,ø,"(t)° 1[ø,ø+")(t)eg¢(ø)

∞

∞

4

l2(H)dt

=K

"

Z ø+"

ø

E∞

∞

eg¢(t)° eg¢(ø)∞

∞

4

l2(H)dt.

From the Lebesgue diÆerentiation theorem, we have for each X 2 L4F1

(≠; l2(H)),

lim"#0

1

"

Z ø+"

ø

E∞

∞

eg¢(t)°X∞

∞

4

l2(H)dt = E

∞

∞

eg¢(ø)°X∞

∞

4

l2(H), for a.e. ø 2 [0, 1).

19

Since L4F1

(≠; l2(H)) is separable, let X run through a countable density subset Qin L4

F1(≠; l2(H)), and denote

E :=[

EX :=[

©

ø : the above relation does not hold for X™

.

Then we have meas(E) = 0. For arbitrary positive ¥, take an X 2 Q such that

E∞

∞

eg¢(ø)°X∞

∞

4

l2(H)< ¥,

then for each ø 2 [0, 1)\E,

lim"#0

1

"

Z ø+"

ø

E∞

∞

eg¢(t)° eg¢(ø)∞

∞

4

l2(H)dt

∑ lim"#0

8

"

Z ø+"

ø

E∞

∞

eg¢(t)°X∞

∞

4

l2(H)dt + 8E

∞

∞

eg¢(ø)°X∞

∞

4

l2(H)

∑ 16E∞

∞

eg¢(ø)°X∞

∞

4

l2(H)< 16¥.

From the arbitrariness of ¥, we conclude this lemma.

From the the above lemma, we have

"°1EZ 1

0

hx1(t), G(t)x1(t)iH dt + "°1E hx1(1), ™x1(1)iH

= o(1) + EZ 1

ø

hyø,"(t), G(t)yø,"(t)iH dt + E hyø,"(1), ™yø,"(1)iH , 8ø 2 [0, 1)\E.

Keeping in mind the above relation, and applying Proposition 3.5, we conclude foreach ø 2 [0, 1)\E,

E≠

eg¢(ø), eP (ø)eg¢(øÆ

l2(H)

= lim"#0

"°1

Ω

EZ 1

0

hx1(t), G(t)x1(t)iH dt + E hx1(1), ™x1(1)iHæ

.

Now fix a predictable version eH¢(·) of H¢(·). Then from (5.2), the aboverelation and the Lebesgue diÆerentiation theorem, we have that for any admissiblecontrol u(·),

0 ∑ E eH¢(ø) +1

2E

≠

eg¢(ø), eP (ø)eg¢(ø)Æ

l2(H), a.e. ø 2 [0, 1).

Integrating ø on [0, 1], we have

0 ∑Z 1

0

Eh

eH¢(ø) +1

2

≠

eg¢(ø), eP (ø)eg¢(ø)Æ

l2(H)

i

dø,

20

which along with Corollary 4.3 and the Fubini theorem, we get

0 ∑ EZ 1

0

h

H¢(ø) +1

2

≠

g¢(ø), P (ø)g¢(ø)Æ

l2(H)

i

dø,

Therefore, the desired variational inequality (2.6) follows from a standard argu-ment. This completes the proof of the stochastic maximum principle.

Acknowledgements

The authors wish to thank Prof. Shanjian Tang for helpful discussions and sugges-tions. The previous version of the paper was reported when Qingxin Meng visitedSichuan University. He is grateful to Prof. Xu Zhang and Dr. Qi Lu for warmhospitality and helpful comments.

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[3] A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter. Representation and controlof infinite dimensional systems. Springer, 1993.

[4] J.M. Bismut. An introductory approach to duality in optimal stochastic control.SIAM review, 20(1):62–78, 1978.

[5] G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions. Cam-bridge University Press, Cambridge, 1992.

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