Adapted solution of a backward semilinear stochastic evolution equation

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Adapted solution of a backward semilinear stochasticevolution equationYing Hu a b & Shige Peng ca Math,CaseH,Universitéde, Cedex3, Provence, Marseille, 13331, Franceb Institut de Math, Université Fudan, R.P de, Fudan, Shanghai, 200433, Chinec Universitéde Shandong, R.P de, Jinan, Shandong, 250100, ChinePublished online: 02 May 2007.

To cite this article: Ying Hu & Shige Peng (1991): Adapted solution of a backward semilinear stochastic evolution equation,Stochastic Analysis and Applications, 9:4, 445-459

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STOCHASTIC ANALYSIS AND APPLICATIONS, 9 ( 4 ) , 445-459 ( 1 9 9 1 )

Adapted Solution of a Backward Semilinear Stochastic Evolution Equation

Ying HU Math,CaseH,Universitk de Provence,

13331Marseille,Cedex3,France Institut de Math,Universit6 Fudan,

200433Shanghai,R.P.de Chine Shige PENG

Institut de Math,Universiti! de Shandong, Jinan, 250100Shandong, R.P.de Chine

Keywords : Backward semilinear stochastic evolution equation, Adapted process, Extended martingale representation theorem.

Abstract : Let It' and H be two separable Hilbert spaces and {W(t) , t E [O, TI) be a cylindrical Wiener process with values in Ii' defined on a probability space ( R , F , P ) , and {Ft) denote its natural filtration. Given X E L2(0, FT, P; H ) , we look for an adapted pair of process {x(t), y(t) : t E [O, TI) with values in H and L2(Ii'; H) respectively (L2(It'; H) is defined in §I), which solves a semilinear stochastic evolution equation of the backward form :

where A is the infinitesimal generator of a Co-semigroup {eAt) on H. The precise meaning of the equation is

Copyright O 1991 by Marcel Dekker, Inc.

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446 HU AND PENG

A linearized version of that equation appears in infinite-dimensional stochastic optimal control theory as the equation satisfied by the adjoint process. We also give our results to the following backward stochastic partial differential equation:

1 Introduction and Notation

1.1 stochastic integral in Hilbert space

Let (JZ,F, P) be a probability space, equipped with filtration Ft. We denote their norms by / . 1, and their scalar products by (.).

A stochastic linear function on a separable Hilbert space I( is a linear mapping from IC to L O ( R , 3, P)(cf. Bensoussan [2]).

Definition 1.1 We say that { W ( t ) , t E R+} is a cylindrical Wiener process on a separable Hilbert space I<, if

(1) W E R+, W ( t ) is a stochastic linear function on I( , (2) V n E N , Q h l , h z , . . . , h n E It',{(W(t)h,,...,W(t)h,),t E R+) is a

Wiener process (not necessary standard) with values in Rn, (3) Vhl , h2 E IC, Qt E R t ,

E ( W ( t ) h l ) ( W ( t ) h Z ) = t (h1, h2)

We assume Ft = u ( W ( s ) , s 5 t )

For any Hilbert space H I , we denote by L$(O,T; H I ) the set of all Ft- progressively measurable processes with values in H I , such that

Obviously L$(O, T; H I ) is a Hilbert space. We can define stochastic integral for x( . ) E L$(O, T ; H ) (c.f. [17] [13])

where { W l ( s ) , s E [0, TI) is a real Wiener process on (R, F, P). Next we consider the space C2(I<; H ) which is the set of Hilbert-Schmidt

operators from K into H, i.e.

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BACKWARD STOCHASTIC EQUATION 44 7

where {en);==, is a orthonormal basis of Ii'. C2(K; H ) is a Hilbert space, and its norm is still denoted by I . 1.

For any $(.) E L$(O, T; L2(Ii'; H ) ) , we can define the stochastic integral loT $(t)dW(t) : L$(O, T; C2(Ii'; H) ) - L2(R, 3 , P; H ) as follows :

The above definition has meaning, since(c.f. [13])

1.2 Backward semilinear stochastic evolution equation

The equation for the adjoint process in infinite-dimensional optimal stochastic control (see Bensoussan [3], Hu and Peng [9]) is a linear version of the following equation

where I< and H are two separable Hilbert spaces and {W(t),t E [O,T]) is a cylindrical Wiener process with values in It' defined on a probability space ( R , 3 , P ) , and { F t , t E [O,T]) is its natural filtration. (i.e. Ft = a(W(s ) , s E [0, t]) ),X E L2(R, FT, P; H), A is the infinitesimal generator of a Go-semigroup {eAt) on H .

f maps R x [0, T] x H x C2(Ii'; H ) into H , f is assumed to be 7J @I

/3(H) @ /3(L2(I(; H)) /P(H) measurable, where 7J denotes the o-algebra of Ft-processively measurable subsets of R x [0, TI, g maps R x [0, T] x H into CZ(K; H), g is assumed to be 7J@/3(H)/P(L2(K; H)) measurable, moreover we assume that f is uniformly Lipschitz with respect to both x and y, g is uniformly Lipschitz with respect to x. We are looking for a pair {zit), y(t); t E [0, TI) with values in H x L2(Ii'; H ) which we require to be {Ft}-adapted. Note that it is the freedom of choosing the process { Y ( ~ ) ) ~ ~ [ O , ~ ~ which will allow us to find an adapted solution.

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448 HU AND PENG

Our main result will be an existence and uniqueness result for an adapted pair {x(t), y(t); t E [O,T]) which solves(1).

Remark 1.1 We can also consider the more general equation :

g maps R x [O,T] x H x L2(Ii'; H) into L2(Ii'; H ) and is P €3 P ( H ) @

P(L2(Ii'; H))/P(L2(It'; H) ) measurable, and satisfies a rather restrictive as- sumtion. (c.f.[14])

In finite dimensional spaces, the above equations(l),(2) were discussed by Pardoux and Peng [14], but their method depends heavily on the Ito formula and cannot be applied directly to our situation. We shall solve this problem with the help of the stochastic Fubini theorem and an extended martingale representation theorem. Note that our method can be applied to more general equation such as :

where G : { ( s , t ) IT 2 s 2 t 2 0) - L ( H ) is continuous, and

We shall also consider the following backward stochastic partial differential equation:

where the conditions on A(.), fi(., .), f (., ., .),g(., .) are given in $4. Since in this case we can use the It6 formula, the proof is somewhat like the finite- dimensional case, we shall only state the results here.

Finally, we would like to indicate that our results here were found to be useful in discussing the so-called stochastic Hamilton-Jacobi-Bellman equations. (c.f.[15])

The paper is organised as follows: in 52, we study a version of equation(1) where f and g do not depend on x. In 53, we study equation(1). And in 54, we study briefly the backward stochastic partial differtial equation. D

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BACKWARD STOCHASTIC EQUATION 449

2 A simplified version of equation(1)

As a preparation for the study of equation(l), we consider in this section simplified versions of that equation. Let us first prove a lemma which plays an important role in our paper.

Lemma 2.1 Iff E L$(O,T; H ) , X E L2(0 ,FT ,P ; H ) , then there exists a unique pair (x, y) E L$(O, T; H ) x Lg(0, T; C2(Ii'; H ) ) satisfies

furthermore, Vt E [0, T]

where

Proof: (1) Uniqueness. We only need to show that if (x, y) E Lg(0, T; H ) x

L$(O, T; C2(Ii'; H ) ) satisfies

x(t) + JT el('-')y(s)dw(s) = 0, t E [0, TI

then x = o , y = o (7)

This is very obvious, we just take E(.JFt) in (6), and get x ( t ) = 0, t E [0, TI; y = 0 follows obviously.

(2) Existence: Put

In order to get the existence of y, we have to use an extended martingale representation theorem and a stochastic Fubini theorem.

Denote E ~ X ~ = E(X1 I z ) , VX, E L1(O, FT, P; H )

From extended martingale representation theorem ([9] [17] ), we know that Vs E [0, TI, there exists uniquely K(s , a ) E Lg(0, T; L2(I<; H) ) , such that

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4 50 HU AND PENG

E" f ( s ) = E f ( s ) + it Ii'(s, B)dW(B), Vt E [ O , TI ( 9 )

Also there exists L( . ) E L;(O, T ; L 2 ( K ; H ) ) , such that

E"X = E X + L(O)dW(O),Vt E [O, T ] (10)

Note that the mapping y: ( s , Ii") E [O,T] x Lg (0 ,T ;L2( I i ' ; H ) ) ---i ( s , JOT K 1 ( 0 ) d W ( 6 ) ) E [0, TI x L i ( R , F T , P ; H ) is a continuous bijection, where LZ(fl, FT, P ; H ) = { X E L2( f l , FT, P ; H ) I E X = 0 ) . And also the projection P,(s, I( ') = Ii" from 10, T ] x Lg (0 ,T ; L2(Ii'; H ) ) -+ Lg(0 , T ; L2(Ii'; H ) ) is continuous. From

we can easily deduce that It' is P[O, TI @ P measurable. Note also from ( 9 ) we can easily deduce that V s E [0, T ]

and that

E lT 1' IIi'(s, 0 ) I2dOds 5 4~ lT / f ( s ) I2ds

From (8) , (9) , (10) , we would have

T x ( t ) = e A ( T - t ) ( ~ - J L(O)dW(O)) - lT e ~ ( ' - ~ ) ( ( s ) - J 3 K ( s , ~ ) d W ( ~ ) ) d s

(12) Because Ii' is P[O, TI @ P measurable and ( l l ) , from the stochastic Fubini

theorem and (12)

where we set

T

y(s ) = e A ( ' - ' ) ~ ( s ) - J ,9 e A ( a - 3 ) ~ ~ ( u 1 s ) d o (14)

Now ( x , Y ) E L$(O, T ; H ) x L$-(0, T ; L 2 ( K ; H ) ) is a solution of (3 ) . The existence is proved.

(3) From (9) , (10) , we know VO 5 t 5 s 5 T ,

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BACKWARD STOCHASTIC EQUATION

From(l4) we know

Now (5) is proved. From (8), we can easily deduce (4).

Corollary 2.1 Given X E LZ(R, FT, P; H ) , ( f ,g) E Lg(0, T ; H ) x Lg(0, T; L2(I<; H)) , there exists a unique pair (x, y) E L%(O, T; H ) x Lg(0, T; C2(I<; H)) set .

We now consider the equation:

where now f : Q x [0, T] x LZ(I<; H ) --+ H is P @ P(L2(1<; H) ) /P (H) measurable ( remember that 7' denote the a-algebra of progressively measurable subsets of R x [0, TI) with the property that

f ( . I 0) E L 2 0 , T; H ) (19)

and there exists c > 0 s.t.

for any yl, yz E L2(I<; H), and (w, t ) a.e. Note that (19)+(20) imply that f (., y(.)) E Lg(0, T ; H ) whenever y E L:(O, T ; L2(I<; H ) ) .

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452 HU AND PENG

Proposition 2.1 Let X E L2(R, FT, P; H ) , g E L'$(O, T; L2(K; H ) ) and f : R x [0, T] x L2(I(; H ) ---( H be a mapping satisfying the above requirements, in particular (19), (20). then there ezists a unique pair (x, y) E L'$(O, T ; H)) x L'$(O, T; L2(I(; H) ) which satisfies (18).

Proof: Uniqueness. Let (xl, yl) and (x2, y2) be two solutions, then

Following from Lemma2.1, we derive Vt E [T - 7, T]

Set 7 = &, then for a.e. s E [T - 7, TI,

From (21), we know Vt 't [T - 27, T - 71

Also use Lemrna2.1 and we can deduce that'for a.e. s E [T - 27, T - 71

thus we can prove the uniqueness of (18).

Existence: With the help of Corollary2.1, we define an approximating sequence by a kind of Picard iteration. Let yo(t) = 0, and {(x,(t), y,(t)) :

t E [0, T])n21 be a sequence in L'$(O, T; H)) x L$(o, T; L2(K; H)) defined recursively by

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From (25) we know

Using again Lemma2.1, we get (set q = &),Vt E [T - q, T]

Since the square roots of the right hand sides of (29) and (30) are summable series, we have that {xn)( resp. {y,)) is a Cauchy sequence in CF(T - 7, T; L2(R, 3, P; H)) (resp. in L$(T - 7,T; C2(K; H)) and passing to the limit in (25) as n + +m, we obtain that the pair (x, y) E CF(T - 7, T; L2(R, 3, P; H)) x L$(T - 7, T; C2(K; H)) defined by

x = lim x,,y = lim yn n-r+ca n- tw

solves the equation (18) for t E [T - q, TI. Now for t E [T - 271, T - q], (26) can be rewritten as

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HU AND PENG

Again use Lemma2.1, we derive V E [T - 27, T - 71

From (32),(33) we know that {x,}( resp. {y,}) is a Cauchy sequence in C F ( T - 217, T - 11 ; L 2 ( R , F, P; H ) ) (resp. in L&(T - 27, T - 17; L2(I<; H ) ) and passing t o the limit in (25) as n ---+ +m, we obtain that t he pair ( x , y) E C F ( T - 277, T ; L 2 ( R , 3, P; H ) ) x L$(T - 27, T ; L2 ( I ( ; H ) ) defined by

x = lim x,, y = l im y, n-+m n-+m

solves the equation ( 1 8 ) for t E [T - 27, TI. Because 17 is fixed, we can deduce the existence o f ( 1 8 ) as above for

t E [O,T].

Remark 2.1 In the proof of uniqueness, because we cannot use the Ito formula for Ixl( t ) - x2 ( t ) l z , we use Lemma2.1 instead to estimate E l x l ( t ) - z2( t ) I2 and E J: l y l ( s ) - y2 ( s ) / 2ds . That is the main idea of our paper. The proof of the existence goes the same way. Note also we have to consider the case first in [T - q , T ] , then extending the result to [O,T]. The rest of our paper is very close to the proof in [Id].

W e can now study the equation:

where f maps R x (0, T ] x H x L2(I<; H ) into H I f is assumed t o be ?J @ P ( H ) @P(L2(I<; H ) ) / P ( H ) measurable, g maps R x [0, TI x H into L2 ( I ( ; H ) , g is assumed t o be P @ , f?(H)/P(C2(K; H ) ) measurable; with the properties that

f (., 0,O) E L:(O, T ; H ) , g( . , 0) E L ~ o , T ; L Z ( K H ) ) (35)

and there exists c > 0 , s.t.

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BACKWARD S T O C H A S T I C EQUATION 455

V X ~ , X ~ E H, y1,y2 E C2(Ii'; H), (w,t) a.e. Note also that (35)+(36)+(37) imply that f (., x(.), y(.)) E L$(O, T; H ) and g(., x(.)) E L$(O, T; C2(Ii'; H) ) whenever x E L:(O, T ; H ) , y E L$(O, T ; C2(I(; H) ) .

Theorem 3.1 Given X E L2(R, FT, P ; H), f and g given as above and satisfying in particular(35)-(37), then there exists a unique pair (x, y) E Lg(0, T; H ) x L$(O, T ; L2(I(; H ) ) which solves equation (34).

Proof: Uniqueness. Let (xl , yl) and ($2, y2) be two solutions in L$(O, T ; H ) x L$(O, T ; C2(Ii'; H) ) . By a similar argument as that of Proposition2.1, (Again use Lemma2.1), then Vt E [T - 7, TI (Set 17 = &)

T E lT yl(s ) - Y Z ( S ) ~ ~ ~ S C (4c2 + 1 ) ~ j Ixl(s) - x2(3)I2d~ (39)

t

From ,(38),(39) we can easily deduce that for a.e.s E [T - 7, TI,

The rest of the proof for uniqueness is the same as that of Proposition2.1 with minor changes.

Existence: We are now going to construct an approximating sequence using a Picard type iteration with the help of Proposition2.1. Let xo(t) = 0, and {(xn(t), yn(t)); t E (0, T])n21 be a sequence in L$(O, T ; H) x L$(O, T; t 2 ( K ; H)) defined recursively by

Using a by now usual procedure, we obtain Vt E [T-7, TI, (set 17 = E Z X 7 v )

From (41),(42) we derive (with I(1 = 9)

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456 HU AND PENG

Iterating that inequality, we obtain

This, together with (42) imples that { x n ) is a Cauchy sequence in C F ( T - 9 , T ; t 2 ( R , 3, P ; H ) ) , { y n } is a Cauchy sequence in L$(T - 17, T ; L2(IC; H ) )

Then, it follows from (40) , that

x = lim xn,y = lim yn n-M n-m

solves equation (34) in [T - p , T ]

Now (40) can be riwritten as: for t E [T - 217, T - 771

I ) - xn(t) + lT-' e ~ ( ' - ~ ) ( f ( Q , x n ( s ) , yn+l(s)) - f ( s , xn - l ( s ) , ~ n ( s ) ) ) d ~

E lT-' l ~ n + ~ ( s ) - Yn(s)l2ds 6 ( 4 2 + l ) ~ l ~ - ~ 1xn(s) - ~ n - i ( s ) l ~ d s

+ 32M2Elxn+l(T - 7 ) - xn(T - p)12 (47)

From (46),(47),(43), we know that Vt E [T - 217, T ]

thus the rest of the proof for the existence is the same as that of Proposi- tion2.1.

4 The Backward Stochastic Partial Differ- t ial Equation

Let V , H be two separable Hilbert spaces such that , V is included and dense in H, the injection being continuous. We identify H with its dual space, and denote by V' the dual of V. We have then

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BACKWARD STOCHASTIC EQUATION 45 7

We will denote by ) I . 1 1 , I . 1, ) I . 1 1 , the norms in V , H , V ' respectively; by < ., . > the duality product between V' and V and by ( a , .) the scalar product in H .

Let the following functions be given:

A(w, t ) : R x [O, TI + L ( V ; V ' ) ,

f l ( w , t , x ) : R x [O,T] x H + V ' ,

f ( w , t , x , y) : R x (0, T ] x V x L 2 ( K ; H ) + H ,

g ( w , t , x ) : R x [ O , T ] x H + L2(I(; H ) , X ( w ) : R + H

A is assumed to be bounded and P measurable, and there exist a > 0 and X E R such that, Vx E V , a.e.t E [0, TI, a s .

fl is assumed to be P 8 P ( H ) measurable, f i ( . , O ) E L>(O, T ; V ' ) and satisfies:

where cl > 0 is a constant. f is assumed to be P @ P ( V ) @ P ( L 2 ( K ; H ) ) measurable, f (., 0,O) E

L$(O, T ; H ) and there exists a constant c2 > 0 such that V x l , 2 2 E V , y l , y2 E

L2(Ii'; H ) )

g is assumed to be P @ @ ( H ) measurable; with the properties that g( . , 0 ) E Lg(0 , T ; L2(It '; H ) ) and there exists a constant c3 > 0 s.t. V x l , x2 E H

Theorem 4.1 Given X E L 2 ( R , F T , P; H ) , f l , f and g given as above and satisfying particular (48)-(51), then there exists a unique pair ( x , y) E L>(O, T ; V ) x L%(O, T ; L2(It '; H ) ) which solves the following equation:

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458 HU AND PENG

In order to prove Theorem4.1, we need the following lemma:

Lemma 4.1 If f E L$(O, T; V1) ,X E L2(R, FT, P; H), then there exists a unique pair (x, y ) E L$(O, T ; V) x L$(O, T; C2(It'; H)) satisfies

We can use the method of finite-dimensional approximation to prove this lemma. See details in [4].

Proof of Theorem4.1: In this case we can use the It6 formula, the proof can be given as in the finite-dimensional case. See the details in [14].

Acknowledgement: The authors would like to thank Professor E. Par- doux for valuable discussions and kind suggestions.

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[ll] Karatzas, S.Shreve, Brownian Motion and Stochastic Calculas, Spinger- Verlag, 1988.

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[17] M.Yor, Existence et unicitC de diffusions & valeurs dans un espace de Hilbert, Ann. Inst. Henri Poincark, lO(55-88), 1974.

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Adapted solution of a backward semilinear stochastic evolution equation