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ITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania

Stochastic variational inequalities innon-convex domains

Aurel R�ascanuDepartment of Mathematics, "Alexandru Ioan Cuza" University, Bd. Carol no. 9-11

&

"Octav Mayer" Mathematics Institute of the Romanian Academy, Bd. Carol I, no.8,

Iasi, Romania

e-mail: [email protected]

Stochastic variational inequalities in non-convex domains 2

ObjectWe give an existence and uniqueness result on the SVI

8<: Xt +Kt = � +

Z t

0

F (s;Xs) ds +

Z t

0

G (s;Xs) dBs; t � 0;

dKt (!) 2 @�' (Xt (!)) (dt) ;

where @�' is the Fréchet subdifferential of a proper l.s.c. semiconvex function ' : Rd !] �1;+1]and Dom (') def=

�x 2 Rd : ' (x) < +1

(in general a non-convex domain).



F Variational Inequalities (VI)

(A) convex domain: completely solved (J. Lions, H. Brezis, V. Barbu, etc. etc.)(B) non-convex domain: A. Marino & M. Tosques (1990), Riccarda Rossi & Giuseppe

Savaré (2004)

F Stochastic Variational Inequalities (SVI)

(A) convex domain: (A.Rascanu (1981,1982, 1996), Ioan Asiminoaei & Aurel Rascanu(1997), Alain Bensoussan & Aurel Rascanu (1994, 1997), El Hassan Es-Saky& Youssef Ouknine(2002), Modeste N'zi &Youssef Ouknine (1997), Etienne Pardoux & Aurel R�ascanu (1998, 1999),U. Haussman & E. Pardoux (1987), A.Z�alinescu (2003) etc

(B) non-convex domain:(�) re�ected diffusions: Pierre-Louis Lions & Alain-Sol Sznitman (1984), Paul Dupuis

& Hitoshi Ishii (1993), Yasumasa Saisho (1987), etc.(�) SVI: : OPEN PROBLEM !!!



Outline

F Preliminaries

F Stochastic variational inequalities in non-convex domainsN Existence and uniqueness resultN Markov solutions of re�ected SDEs.

F AnnexN Classical Skorohod problemN Generalized Skorohod equationsN A forward stochastic inequalityN Tightness and some Itô's integral properties

F References



1 PreliminariesLet E be a nonempty closed subset of Rd and NE (x) be the closed external normal cone of E at x 2 Ei.e.

NE (x)def=

�u 2 Rd : lim

�&0

dE (x + �u)

�= juj

�;

De�nition 1 We say that E satis�es the uniform exterior ball condition, abbreviated UEBC, if

� NE (x) 6= f0g for all x 2 Bd (E) ;

� there exists r0 > 0; such that 8 x 2 Bd (E) and 8 u 2 NE (x), juj = r0 it holds:

dE (x + u) = r0 (or equivalent B (x + u; r0) \ E = ;)

(we say that E satis�es the r0 � UEBC).

We have the following equivalence:

Lemma 1 The following assertions are equivalent:

(i) E satis�es the uniform exterior ball condition;ITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


(ii) 9 > 0 and for all x 2 Bd (E) there exists x 2 Rdn f0g such that

hx; y � xi � jxj jy � xj2 ; 8y 2 E:

Lemma 2 If E satis�es the r0 � UEBC; then

NE (x) =

�x : hx; y � xi � 1

2r0jxj jy � xj2 ; 8y 2 E

�;

Denote Ec = Rd n E and the "�interior of E the set

E" = fx 2 E : dist (x;Ec) � "g = fy : B (y; ") � Eg:

De�nition 2 E � Rd satis�es the shifted uniform interior ball condition (SUIBC) if there exist �; � > 0,and for every y 2 E there exist �y 2]0; 1] and vy 2 Rd; jvyj � 1 such that

(SUIBC)

((i) �y � (jvyj + �y)2 2 � �;(ii) B (x + vy; �y) � E; 8 x 2 E \B (y; �)

(1)

Note that (SUIBC) implies that int (E) 6= ;:ITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


Let x; v 2 Rd, r > 0:We call the set

Dx (v; r) = conv�x;B (x + v; r)

=�x + t (u� x) : u 2 B (x + v; r) ; t 2 [0; 1]

a (jvj ; r)�drop (or a comet) of vertex x and running direction v.

Proposition 1 Let E = E � Rd: If int (E) 6= ;: Each of the following conditions implies that E satis�esthe shifted uniform interior ball condition (SUIBC).

(A1)

lim"&0

1p"supx2E

dist (x;E") = 0

(A2) (uniform linear cover) There exists N � 1 such that for all " > 0 :

E �[y2E"

B (y; "N) :

(A3) (uniform interior (h0; r0)�drop condition) There exist 0 < r0 � h0 and for all x 2 E there existsITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


vx 2 Rd, jvxj = h0, such that:

Dx (vx; r0) � E

(A4) (uniform interior ball condition) There exists r0 > 0 such that Ec satis�es the the r0� uniform exteriorball condition.

Example 1 Let E � Rd de�ned by8>><>>:(i) E =

�x 2 Rd : � (x) � 0

; where � 2 C2b

�Rd�;

(ii) int (E) =�x 2 Rd : � (x) < 0

;

(iii) Bd (E) =�x 2 Rd : � (x) = 0

and jr� (x)j = 1 8 x 2 Bd (E) :

(2)

The set E satis�es UEBC:the uniform exterior ball condition and SUIBC: the shifted uniform interiorball condition.

Proof. Note that at any boundary point x 2 Bd (E) ; r� (x) is a unit normal vector to the boundary ,pointing towards the exterior of E. Hence for all x 2 Bd (E) we have NE (x) = f� r� (x) : � � 0g andNEc (x) = f��r� (x) : � � 0g :ITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


Since for all y 2 E and x 2 Bd (E) ;

hr� (x) ; y � xi = � (y)� � (x)�Z 1

0

hr� (x + � (y � x))�r� (x) ; y � xi d�

�M jy � xj2

then by Lemma 1 E satis�es (UEBC).If y 2 Ec and x 2 Bd (E) ; then � (y) > 0, � (x) = 0 and

h�r� (x) ; y � xi = �� (y) +Z 1

0

hr� (x + � (y � x))�r� (x) ; y � xi d�

�M jy � xj2

that is Ec satis�es (UEBC) and consequently, by Proposition 1, E satis�es (SUIBC).

A function ' : Rd ! ]�1;+1] is proper if

Dom (')def=�v 2 Rd : ' (v) < +1

6= ;

and Dom (') has no isolated point.

De�nition 3 The Fréchet subdifferential of ' at x 2 Rd is de�ned by @�' (x) = ;; if x =2 Dom (') andITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


for x 2 Dom (') ;

@�' (x) = fx 2 Rd : lim infy!x

' (y)� ' (x)� hx; y � xijy � xj � 0g:;

the domain of the multivalued operator @�' : Rd� Rd is de�ned by

Dom�@�'

� def=�x 2 Rd : @�' (x) 6= ;

If E is a nonempty closed subset of Rd and

' (x) = IE (x) =

(0; if x 2 E;+1; if x =2 E;

then ' is l.s.c. and

@�IE (x) = fx 2 Rd : lim supy!x; y2E

hx; y � xijy � xj � 0g

is the Fréchet normal cone at E in x: By a result of Colombo&Goncharov (2001) we have for anyclosed subset E of a Hilbert space

@�IE (x) = NE (x)



De�nition 4 ' : Rd ! ]�1;+1] is a semiconvex function if

(i) Dom (') satis�es the UEBC ;

(ii) Dom (@�') 6= ;

(iii) there exist 1; 2 � 0 such that for all (x; x) 2 @�', y 2 Rd :

hx; y � xi + ' (x) � ' (y) + ( 1 + 2 jxj) jy � xj2 :

We also say that ' is a ( 1; 2)�semiconvex function.

Remark that� If E satis�es r0 � UEBC then ' = IE is ( 1; 2)�semiconvex l.s.c. function for all 1 � 0 and 2 �

1

2r0.

� A convex function ' is ( 1; 2)�semiconvex function for all 1; 2 � 0:If ' : Rd ! ]�1;+1] is a semiconvex function, then there exists a 2 R such that

' (y) + a jyj2 + a � 0 for all y 2 Rd:

Example 2 If E is a closed bounded subset of Rd satisfying the UEBC and g 2 C2�Rd�(or g 2 C

�Rd�

is a convex function), then f : Rd !]�1;+1], f (x) = IE (x) + g (x) is a l.s.c. semiconvex function.ITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


2 Stochastic variational inequalities in non-convex domains

2.1 Existence and uniqueness resultConsider the following SVI (also called generalized stochastic Skorohod equation

8<: Xt +Kt = � +

Z t

0

F (s;Xs) ds +

Z t

0

G (s;Xs) dBs; t � 0;

dKt (!) 2 @�' (Xt (!)) (dt) ;(3)

where we haveN fBt : t � 0g is aRk�valued Brownian motion with respect to the given stochastic basis (;F ;P; fFtgt�0) ;

N Carathéodory conditions: F (�; �; �) : � [0;+1[� Rd ! Rd and G (�; �; �) : � [0;+1[� Rd !Rd�k are

�P ;Rd

��Carathéodory functions that is((a) F (�; �; x) and G (�; �; x) are p.m.s.p., 8 x 2 Rd;(b) F (!; t; �) and G (!; t; �) are continuous function dP dt� a:e:

(4)



Denoting

F# (s) = supnjF (t; u)j : u 2 Dom (')

o;

G# (s) = supnjG(t; u)j : u 2 Dom (')

owe assume that the following are satis�edN Boundedness conditions : for all T � 0 :8>>><>>>:

(a)

Z T

0

F# (s) ds <1; P� a.s.

(b)

Z T

0

��G# (s)��2 ds <1; P� a.s. (5)

N Monotonicity and Lipschitz conditions : there exist � 2 L1loc (0;1) and ` 2 L2loc (0;1;R+) ;such that dP dt� a:e: :8>>>><>>>>:

(MF ) Monotonicity condition :hx� y; F (t; x)� F (t; y)i � � (t) jx� yj2; 8x; y 2 Rd,

(LG) Lipschitz condition :jG(t; x)�G(t; y)j � ` (t) jx� yj; 8x; y 2 Rd.

(6)



N @�' is the Fréchet subdifferential of a proper l.s.c. semiconvex function ' : Rd !]�1;+1] andDom (')

def=�x 2 Rd : ' (x) < +1

(in general a non-convex domain) has the property (SUIBC) :

shifted uniform interior ball condition . Moreover of satis�es

j' (x)� ' (y)j � L + L jx� yj ; 8 x; y 2 Dom (') :

De�nition 5 (I) Given a stochastic basis (;F ;P;Ft)t�0 and a Rk�valued Ft�Brownian motionfBt : t � 0g ; a pair (X;K) : �[0;1[! Rd�Rd of continuousFt�progressively measurable stochasticprocesses is a strong solution of the SDE (3) if P� a:s: ! 2 :8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

j) Xt 2 Dom ('); 8 t � 0; ' (X�) 2 L1loc (0;1)jj) K� 2 BVloc

�[0;1[ ;Rd

�; K0 = 0,

jjj) Xt +Kt = � +

Z t

0

F (s;Xs) ds +

Z t

0

G (s;Xs) dBs; 8 t � 0;

jv) 8 0 � s � t; 8y : [0;1[! Rd continuous :Z t

s

hy (r)�Xr; dKri +Z t

s

' (Xr) dr

�Z t

s

' (y (r)) dr +

Z t

s

jy (r)�Xrj2 ( 1dr + 2d lKlr)

(7)



that is

(X� (!) ; K� (!)) = SP�@�'; � (!) ;M� (!)

�; P� a:s: ! 2 ;

with

Mt =

Z t

0

F (s;Xs) ds +

Z t

0

G (s;Xs) dBs :

(II) Let F (!; t; x) = f (t; x) ; G (!; t; x) = g (t; x) and � (!) = x0 be independent of !. If thereexists a stochastic basis (;F ;P;Ft)t�0 , a Rk�valued Ft�Brownian motion fBt : t � 0g and a pair(X�; K�) : � [0;1[! Rd � Rd of Ft�p.m.c.s.p. such that

(X� (!) ; K� (!)) = SP�@�';x0;M� (!)

�; P� a:s: ! 2 ;

with

Mt =

Z t

0

f (s;Xs) ds +

Z t

0

g (s;Xs) dBs;

then the collection (;F ;P;Ft; Bt; Xt; Kt)t�0 is called a weak solution of the SDE (3).

We �rst give a uniqueness result for strong solutions.

Proposition 2 (Pathwise uniqueness) . Let (;F ;P;Ft; Bt)t�0 be given and the assumption (28) beITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


satis�ed. Assume that

F (�; �; �) : � [0;+1[� Rd ! Rd and G (�; �; �) : � [0;+1[� Rd ! Rd�k

satisfy (4), (5) and (6). Then the SDE (3) has at most one strong solution.

Proof. Let (X;K) and (X; K) be two solutions corresponding to � and respectively �. Since

dKt 2 @�' (Xt) (dt) and dKt 2 @�'(Xt) (dt)

then, by Lemma ??, for p � 1 and � > 0DXt � Xt; (F (t;Xt) dt� dKt)�

�F (t; Xt)dt� dKt

�E+

�1

2mp + 9p�

� ��G (t;Xt)�G(t; Xt)��2 dt � jXt � Xtj2dVtwith

Vt =

Z t

0

�� (s) ds +

�1

2mp + 9p�

�`2 (s) ds + 2 1ds + 2d lKls + 2d l K ls

�:

Therefore, by Proposition 7 (Annex), we get

Eh1 ^

e�V �X � X� pT

i� Cp;�E

h1 ^

�� pi :ITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


Hence the uniqueness follows.

Lemma 3 (The case of aditive noise) If � 2 L0�;F0;P;Dom (')

�and M 2 S0d , M0 = 0; then there

exists a unique pair (X;K) 2 S0d � S0d strong solution of the problem8<: Xt (!) +Kt (!) = � (!) +

Z t

0

F (!; s;Xs (!)) ds +Mt (!) ; t � 0;

dKt (!) 2 @�' (Xt (!)) (dt) ;

P� a:s: ! 2 ; that is

(X� (!) ; K� (!)) = SP�@�'; � (!) ;M� (!)

�; P� a:s: ! 2 :

To study the general SDE (3) we consider only the case when F;G and � are independent of !: Hencewe have 8<: Xt +Kt = x0 +

Z t

0

f (s;Xs) ds +

Z t

0

g (s;Xs) dBs; t � 0;

dKt (!) 2 @�' (Xt (!)) (dt) ;(8)

where f (�; �) : [0;+1[� Rd ! Rd and g (�; �) : [0;+1[� Rd ! Rd�k .ITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


As above, denote

f# (s)def= sup

njf (t; u)j : u 2 Dom (')

o;

g# (s)def= sup

njg(t; u)j : u 2 Dom (')

o:

Proposition 3 If x0 2 Dom (') andZ T

0

hf# (s)2 + g# (s)4

ids <1; 8 T � 0:

then the problem (8) has a weak solution (;F ;P; ;Ft; Xt; Kt; Bt)t�0 .

Proof. Step 1. Approximating sequence. Let�;F ;P;FBt ; Bt

�t�0 be a Brownian motion. By Lemma 3

there exists a unique pair (Xn; Kn) : � [0;1[! Rd�Rd of FBt �progressively measurable continuousstochastic processes, solution of the approximating problem8<: Xn

t +Knt = x0 +

Z t

0

f�s;Xn

s� 1n

�ds +

Z t

0

g�s;Xn

s� 1n

�dBs; t � 0;

dKnt (!) 2 @�' (Xn

t (!)) (dt) :

(9)



Denote

Mnt =

Z t

0

f�s;Xn

s� 1n

�ds +

Z t

0

g�s;Xn

s� 1n

�dBs

Since

E�sup0��"

jMnt+� �Mn

t j4

�� C

"�Z t+"

t

f# (s) ds

�4+

�Z t+"

t

��g# (s)��2 ds�2#

� "C"supt2[0;T ]

�Z t+"

t

��f# (s)��2 ds�2 + supt2[0;T ]

Z t+"

t

��g# (s)��4 ds#then, by Proposition 8, the family of laws of fMn : n � 1g is tight on C

�[0;1[ ;Rd

�:

Therefore for all T � 0

limN%1

�supn�1

P (kMnkT � N)�= 0

and for all a > 0 and T > 0 :

lim"&0

�supn�1

P�(mMn ("; [0; T ]) � a)

��= 0 (10)



Denoting

�Mn (") = " +mMn ("; [0; T ])

= " + sup fjMnt �Mn

s j : 0 � s � t � T; t� s � "g

we can replace in (10)mMn by �Mn:

Step 2. Tightness. Let T � 0 be arbitrary. We now show that the family of laws of the random variablesUn = (Xn; Kn; lKnl) is tight on X =C

�[0; T ] ;R2d+1

�:

From (30-c) we deduce

mUn ("; [0; T ]) � G (Mn)p�Mn ("); a.s.

where x 7�! G (x) = C�T; kxkT ;��1x

�: C

�[0; T ] ;Rd

�! R+ is bounded on compact subset of

C�[0; T ] ;Rd

�:

This inequality yields that fUn;n 2 N�g is tight on X (recall that U0 = (x0; 0; 0; 0)).Then by the Prohorov theorem there exists a subsequence such that as n!1

(Xn; Kn; lKnl ; B)! (X;K; V;B) in law

on X�C�[0; T ] ;Rk

�: We can and do assume that both the sequence f(Xn; Kn; lKnl ; B) : n 2 N�g

and the limit (X;K; V;B) are de�ned on the same probability space, still denoted (;F ;P) :Step 3. Passing to the limit in (9).



By Proposition 9 we have for all 0 � s � t; P� a:s:

X0 = x0 ; K0 = 0; Xt 2 Dom (');lKlt � lKls � Vt � Vs and 0 = V0 � Vs � Vs

(11)

Since for all 0 � s < t, n 2 N�Z t

s

' (Xn� ) d� �

Z t

s

' (y (� )) d� �Z t

s

hy (� )�Xn� ; dK

n� i

+

Z t

s

jy (� )�Xn� j2 ( 1d� + 2d lKnl� ) ; a:s:

then by Proposition 9 we inferZ t

s

' (X� ) d� �Z t

s

' (y (� )) d� �Z t

s

hy (� )�X� ; dK�i

+

Z t

s

jy (� )�X� j2 ( 1d� + 2dV� )(12)

Now by (11) and an auxiliary result on the inequality (12) we have

dK� 2 @�' (X� ) (d� ) :

Step 3. Passing to the limit in (9-a).



By the Skorohod theorem, we can choose a probability space (;F ;P), and some random triples( �Xn; �Kn; �Bn), ( �X; �K; �B) de�ned on that probability space, having the same laws as resp. (Xn; Kn; B)

and (X;K;B); such that moreover as n!1;

( �Xn; �Kn; �Bn)P�a:s:��! ( �X; �K; �B) in C

�[0; T ] ;R2d+k

�: (13)

By Proposition 11 we have:��Bn; fF �Bn; �Xn

t g�, n � 1, and

��B; fF �B; �X

t g�are Rk�Brownian motions.

Using the Lebesgue theorem and, once again Proposition 11, we infer for n!1

�Mn� = x0 +

Z �

0

f�s; �Xn

s� 1n

�ds +

Z �

0

g�s; �Xn

s� 1n

�d �Bns

�! �M� = x0 +

Z �

0

f�s; �Xs

�ds +

Z �

0

g�s; �Xs

�d �Bs; in S0d [0; T ] :

By Proposition 10 it follows

L��Xn; �Kn; �Bn; �Mn

�= L (Xn; Kn; Bn;Mn) on C

�R+;Rd+d+k+d

�Since for every t � 0

Xnt +K

nt �Mn

t = 0; a:s:;



then by Proposition 9 we have

�Xnt +

�Knt � �Mn

t = 0; a:s:

and letting n!1;�Xt + �Kt � �Mt = 0; a:s: :

that is P� a:s:

�Xt + �Kt = x0 +

Z t

0

f�s; �Xs

�ds +

Z t

0

g�s; �Xs

�d �Bs; 8 t 2 [0; T ] :

Hence��; �F ; �P; �X; �B

�is a weak solution of the SDE (8).

Finally with similar arguments as used by Ykeda-Watanabe weak existence + pathwise uniquenessimplies strong existence. Hence

Theorem 1 Under the assumptions of Proposition 3 the problem (8) has a unique strong solution(Xt; Kt)t�0 :



2.2 Markov solutions of re�ected SDEs.In this subsection we study the Markov property of the solutions of re�ected SDEs. when the set Ehas a particular form. Assume that8>><>>:

(i) E =�x 2 Rd : � (x) � 0

; where � 2 C2b

�Rd�;

(ii) int (E) =�x 2 Rd : � (x) < 0

;

(iii) Bd (E) =�x 2 Rd : � (x) = 0

and jr� (x)j = 1 8 x 2 Bd (E) :

(14)

The set E satis�es (UEBC:the uniform exterior ball condition) and (SUIBC: the shifted uniform interiorball condition)Let t 2 [0;1[ and � 2 L0 (;Ft;P;E) be �xed. Consider the equation8<: X t;�

s +Kt;�s = � +

Z s

t

f�r;X t;�

r

�dr +

Z s

t

g�r;X t;�

r

�dBr; s � t;

dKr (!) 2 @�IE�X t;�r (!)

�(dr) ;

(15)

where f (�; �) : [0;+1[ � Rd ! Rd and g (�; �) : [0;+1[ � Rd ! Rd�k are continuous functions andsatisfy: there exist � 2 R and ` > 0 such that for all u; v 2 Rd

(i) hu� v; f (t; u)� f (t; v)i � �ju� vj2

(ii) jg(t; u)� g(t; v)j � `ju� vj:(16)



If follows from Theorem 1 and Theorem 2 that there exists a unique pair�X t;�; Kt;�

�: � [0;1[ !

Rd � Rd of continuous progressively measurable stochastic processes is a strong solution of the SDE(15) that is P� a:s: ! 2 :8>>>>>>>>>>><>>>>>>>>>>>:

(j) X t;�s 2 E and X t;�

s^t = � for all s � 0(jj) Kt;�

� 2 BVloc�[0;1[ ;Rd

�; Kt;�

s = 0 for all 0 � s � t;

(jjj) X t;�s +Kt;�

s = � +

Z s

t

f�r;X t;�

r

�dr +

Z s

t

g�r;X t;�

r

�dBr; 8 s � t;

(jv)xyKt;�

xys=

Z s

t

1Bd(E)�X t;�r

�dxyKt;�

xyr; 8 s � t;

(v) Kt;�s =

Z s

t

r��X t;�r

�dxyKt;�

xyr; 8 s � t:

(17)

Remark that by Theorem 2 the conditions (jv) & (v) are equivalent to(9 > 0 such that 8 y : [0;1[! E continuous :y (r)�X t;�

r ; dKt;�r

��

��y (r)�X t;�r

��2 dxyKt;�xyr:

If�X t;�; Kt;�

�and (X t;�; Kt;�) are two solutions, then from this last inequality it followsX t;�r �X t;�

r ; dKt;�r � dKt;�

r

�+

��X t;�r �X t;�

r

��2 �dxyKt;�xyr+ d lKt;� lr

�� 0; a:s: (18)



Proposition 4 Let the assumptions (14), 16) be satis�ed and

sup(t;x)2[0;T ]�E

[jf (t; x)j + jg (t; x)j] <1:

Then for all p � 1; � > 0 and s � t;

(j) EFt supr2[t;s]

��X t;�r �X t;�

r

��p � C j� � �jp exp [C (s� t)] ; a.s.,(jj) EFt sup

r2[t;s]

��Kt;�r

��p � EFt xyKt;�xyps� C (1 + (s� t)p) ; a.s.

(jjj) EFt supr2[t;s]

��X t;�r

��p � C (1 + (s� t)p + j�jp) ;(jv) EFte�jKt;�

s j � EFte�lKt;�l

s � exp�C� +

�C� +

C2�2

2

�t

�; a.s..

(19)

where C is a constant independent of �; �; s; t and �:If the monotonicity condition (16-i) is replaced by

jf (t; u)� f (t; v)j � �ju� vj;

and � 2 C3b�Rd�; then we moreover have

EFt supr2[t;s]

��Kt;�r �Kt;�

r

��p + EFt supr2[t;s]

��xyKt;�xyr�xyKt;�

xyr

��p � CeC(s�t) j� � �jp ; (20)



where C is a constant independent of �; �; s; t.

Corollary 1 Let the assumptions of Proposition (4) be satis�ed and E a bounded set. Then for everyT > 0 and p � 1 the mapping

(t; x) 7!�X t;x� ; K

t;x� ;xyKt;x

xy� : [0; T ]� E ! Spd [0; T ]� Spd [0; T ]� S

p1 [0; T ]

is continuous. Moreover if h1; h2 : [0; T ]� E ! R are bounded continuous functions, then

(t; x) 7! EZ T

t

h1(s;Xt;xs )ds + E

Z T

t

h2(s;Xt;xs ) d

xyKt;xxys: [0; T ]� E ! R (21)

is continuous.Moreover

Proposition 5 The solution�X0;xs : s � 0

of the SDE (15) is a strong Markov process with

(i) the transition probability

P (t; x; s;G) = P�X t;xs 2 G

�for t; s � 0 and G 2 Bd ;



(ii) the evolution operator Pt;s : Bb(Rd)! Bb(Rd); 0 � t � s;

(Pt;s') (x) = E'�X t;xs

�;

(iii) the in�nitesimal generator At satisfying

D def=�' 2 C2b

�Rd�: r' (x) = 0 if x 2 Bd (E)

� Dom(At); for all t � 0

and for ' 2 D

At (') (x) =1

2Tr [g (t; x) g� (t; x)'00xx (x)] + hf (t; x) ; '0x (x)i

=1

2

dXi;j=1

(gg�)ij(t; x)@2' (x)

@xi@xj+

dXi=1

fi(t; x)@' (x)

@xi:



3 Annex

3.1 Classical Skorohod problemLet E � Rd be a non empty closed subset of Rd:We assume

(UEBC) E satis�es the uniform exterior ball conditionand

(SUIBC) E satis�es the shifted uniform interior ball condition

De�nition 6 Let E satisfy the r0 � UEB condition. A pair (x; k) is a solution of the Skorohod problem(and we write (x; k) = SP (E;x0;m)) if� x; k : [0;1[! Rd are continuous functions and

� for all 0 � s � t � T :8>>>>>>><>>>>>>>:

j) x (t) 2 E;jj) k 2 BVloc

�[0;1[ ;Rd

�; k (0) = 0,

jjj) x (t) + k (t) = x0 +m (t) ; ;

jv) 8 y 2 C (R+;E) :Z t

s

hy (r)� x (r) ; dk (r)i � 1

2r0

Z t

s

jy (r)� x (r)j2 d lklr :

(22)



Theorem 2 Let x0 2 E and m : R+ ! Rd a continuous function such that m (0) = 0: Under theassumptions (UEBC) and (SUIBC) the Skorohod problem (22) has a unique solution. Moreover(22� jv), (23), (24) ; where8>>>>><>>>>>:

(a) lklt =Z t

0

1x(s)2Bd(E)d lkls ;

(b)k (t) =

Z t

0

nx(s)d lkls ; where nx(s) 2 NE (x (s))

and��nx(s)�� = 1; d l k ls �a:e::

(23)

and 8<:9 > 0 such that 8 y : [0;1[! E continuous :Z t

s

hy (r)� x (r) ; dk (r)i � Z t

s

jy (r)� x (r)j2 d lklr :(24)



3.2 Generalized Skorohod equationsPROBLEM : 9? a pair (x; k) of continuous functions x; k : [0;1[ ! Rd solution of the generalizedSkorohod differential equation8<: x (t) + k (t) = x0 +

Z t

0

f (s; x (s)) ds +m (t) ; t � 0;

dk (t) 2 @�' (x (t)) (dt) ;(25)

By dk (t) 2 @�' (x (t)) (dt) we understand

a) x 2 C�[0;1[ ;Dom (')

�; ' (x) 2 L1loc (0;1) ;

b) k 2 C�[0;1[ ;Rd

� TBVloc

�[0;1[ ;Rd

�; k (0) = 0;

c) 8 0 � s � t; 8y : [0;1[! Rd continuous :Z t

s

hy (r)� x (r) ; dk (r)i +Z t

s

' (x (r)) dr

�Z t

s

' (y (r)) dr +

Z t

s

jy (r)� x (r)j2 ( 1dr + 2d lklr)

(26)



Assumptions:

(A1) ((i) x0 2 Dom (')(ii) m 2 C

�[0; T ] ;Rd

�; m (0) = 0;

(27)

(A2)

(i) ' : Rd ! ]�1;+1] is proper l.s.c. ( 1; 2)� semiconvex function;(ii) j' (x)� ' (y)j � L + L jx� yj ; 8 x; y 2 Dom (')(iii) Dom (') satis�es shifted uniform interior ball condition (SUIBC);

(28)

(A3)

� the function f (�; �) : [0;+1[� Rd ! Rd is Carathéodory� there exist � 2 L1loc (0;1) ; such that the monotonicity condition

hx� y; f (t; x)� f (t; y)i � � (t) jx� yj2; 8 x; y 2 Rd

is satis�ed a:e: t � 0� Boundedness : for all T � 0;Z T

0

f# (s) ds <1; where f# (t)def= sup

njf (t; u)j : u 2 Dom (')

oITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


Theorem 3 (Generalized Skorohod Equation) Under the assumptions (A1) ; (A2) ; (A3) the Eq. (25)has a unique solution (x; k) :

The proof is based on the following result:

Proposition 6 (Generalized Skorohod Problem) Let the assumptions (A1) and (A2) be satis�ed.I. 9 CT : C

�[0; T ] : Rd

�! R+ which is bounded on compact subsets of C

�[0; T ] : Rd

�such that for

any solution (x; k) is solution of (x (t) + k (t) = x0 +m (t) ; t � 0;dk (t) 2 @�' (x (t)) (dt) ;

(29)

(we write (x; k) = SP(@�';x0;m)), then

(a) lklT = kkkBV ([0;T ];Rd) � CT (m) ;(b) kxkT = kxkC([0;T ];Rd) � jx0j + CT (m) ;

(c)jx (t)� x (s)j + lklt � lkls � CT (m)�

p�m (t� s);

8 0 � s � t � T:

(30)



If moreover m 2 C�[0; T ] ;Rd

�, x0 2 Dom (') and (x; k) = SP(@�'; x0; m); then

kx� xkT + kk � kkT � �T (m; m)�hjx0 � x0j +

pkm� mkT

i; (31)

II. The problem (29) has a unique solution (x; k) of continuous functions x; k : [0;1[! Rd:

3.3 A forward stochastic inequalityLet X 2 S0d be a continuous local semimartingale of the form

Xt = X0 +Kt +

Z t

0

GsdBs; t � 0; P� a:s:; (32)

and for some p � 1 and � > 1 as signed measures on [0;1[

dDt + hXt; dKti +�1

2mp + 9p�

�jGtj2 dt � 1p�2dRt + jXtjdNt + jXtj2dVt;

where mp = 1 _ (p� 1) and� K 2 S0d; K� 2 BVloc

�[0;1[ ;Rd

�; K0 = 0; P� a:s:;

� G 2 �0d�k;� D;R;Nare P�m.i.c.s.p. D0 = R0 = N0 = V0 = 0;� V is a P�m.local b-v.c.s.p., V0 = 0:ITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


Proposition 7 There exists a constant Cp;� such that: P� a:s:; for all 0 � t � s and � � 0

EFtke�VXkp

[t;s]

(1+�ke�VXk2[t;s])p=2 + EFt

Z s

t

e�pVr jXrjp�2

(1+�je�VrXrj2)(p+2)=2dDr + EFt

�Z s

t

e�2Vr

(1+�je�VrXrj2)2

�dDr + jGrj2 dr

��p=2� Cp;�

�e�pVtjXtjp

(1+�e�2VtjXtj2)p=2 + EFt

�Z s

t

e�2Vr1p�2dRr

�p=2+ EFt

�Z s

t

e�VrdNr

�p�:

(33)

In particular if R = N = 0 then P� a:s:

EFth1 ^

e�VX p[t;s]

i� Cp;�

h1 ^

��e�VtXt��pi ;



3.4 Tightness and some Itô's integral properties

Proposition 8 Let fXnt : t � 0g, n 2 N�, be a family of Rd�valued continuous stochastic processes

de�ned on probability space (;F ;P) : Suppose that for every T � 0; there exist � = �T > 0 andb = bT 2 C (R+) with b(0) = 0; (both independent of n) such that

(j) limN!1

�supn2N�

P(fjXn0 j � Ng)

�= 0;

(jj) E�1 ^ sup

0�s�"jXn

t+s �Xnt j�

�� " � b("); 8 " > 0, n � 1; t 2 [0; T ] ;

Then fXn : n 2 N�g is tight in C(R+;Rd):

Proposition 9 ' : Rd !] � 1;+1] is a l.s.c. function. Let (X;K; V ), (Xn; Kn; V n) ; n 2 N; beC�[0; T ] ;Rd

�2 � C ([0; T ] ;R)�valued random variables, such that(Xn; Kn; V n)

law��!n!1

(X;K; V )



and for all 0 � s < t; and n 2 N�;

lKnlt � lKnls � V nt � V ns a:s:Z t

s

' (Xnr ) dr �

Z t

s

hXnr ; dK

nr i ; a:s:;

then

�(�) lKlt � lKls � Vt � Vs a:s:

(�)

Z t

s

' (Xr) dr �Z t

s

hXr ; dKri ; a:s::

Proposition 10 Let X; X 2 S0d [0; T ] and B; B be two Rk�Brownian motions and g : R+ � Rd ! Rd�kbe a function satisfying

g (�; y) is measurable 8 y 2 Rd; andy 7! g (t; y) is continuous dt� a:e::

If

L (X;B) = L(X; B); on C�R+;Rd+k

�ITN Workshop "Finance and Insurance", March 16-20, 2009, Jena, Germany Aurel R�ascanu, "Alexandru Ioan Cuza" University, Iasi, Romania


then

L�X;B;

Z �

0

g (s;Xs) dBs

�= L

�X; B;

Z �

0

g�s; Xs

�dBs

�; on C

�R+;Rd+k+d

�:

Proposition 11 Let B;Bn; �Bn : � [0;1[! Rk and X;Xn; �Xn : � [0;1[! Rd�k; be c.s.p. such that

(i) Bn is FBn;Xn

t �Brownian motion 8 n � 1;

(ii) L(Xn; Bn) = L��Xn; �Bn

�on C(R+;Rd�k � Rk) for all n � 1;

(iii)

Z T

0

�� Xns � �Xs

��2 ds + supt2[0;T ]

�� Bnt � �Bt�� in probability, as n!1; for all T > 0:

Then��Bn; fF �Bn; �Xn

t g�; n � 1; and

��B; fF �B; �X

t g�are Brownian motions and as n!1

supt2[0;T ]

��Z t

0

�Xns d�Bns �!

Z t

0

�Xsd �Bs

�� ! 0 in probability:



[1] Barbu, Viorel & R�ascanu, Aurel : Parabolic variational inequalities with singular inputs, DifferentialIntegral Equations 10 (1997), no. 1, 67�83.

[2] Bensoussan, Alain & R�ascanu, Aurel : Stochastic variational inequalities in in�nite-dimensionalspaces, Numer. Funct. Anal. Optim. 18 (1997), no. 1-2, 19�54.

[3] Buckdahn, Rainer & R�ascanu, Aurel : On the existence of stochastic optimal control of distributedstate system, Nonlinear Anal. 52 (2003), no. 4, 1153�1184.

[4] Colombo, Giovanni & Goncharov, Vladimir : Variational Inequalities and Regularity Properties ofClosed Sets in Hilbert Spaces, Journal of Convex Analysis, Volume 8 (2001), No. 1, 197-221.

[5] Degiovanni, M. & Marino, A. & Tosques, M. : Evolution equations with lack of convexity, NonlinearAnal. Th. Meth. Appl. 9(2) (1985), 1401�1443.

[6] Dupuis, Paul & Ishii, Hitoshi : SDEs with oblique re�ection on nonsmooth domains, Ann. Probab. 21(1993), no. 1, 554�580.

[7] Lions, Pierre-Louis & Sznitman, Alain-Sol : Stochastic differential equations with re�ecting bound-ary conditions, Comm. Pure Appl. Math. 37 (1984), no. 4, 511�537.

[8] Marino, A. & Tosques, M. : Some variational problems with lack of convexity, Methods of NonconvexAnalysis, Proceedings from CIME, Varenna 1989 , Springer,Berlin (1990)

[9] Pardoux, Etienne & R�ascanu, Aurel : SDEs and Applications (book in preprint)[10] R�ascanu, Aurel : Deterministic and stochastic differential equations in Hilbert spaces involving multi-

valued maximal monotone operators, Panamer. Math. J. 6 (1996), no. 3, 83�119.



Thank you for your attention !


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