SET-VALUED STOCHASTIC INTEGRALS AND STOCHASTIC INCLUSIONS IN A PLANE

Differential Inclusions, Control and Optimization 21 (2001 ) 249–259

SET-VALUED STOCHASTIC INTEGRALS AND STOCHASTIC INCLUSIONS IN A PLANE

W ladys law Sosulski Institute of Mathematics University of Zielona G´ora

Podg´orna 50, 65–246 Zielona G´ora, Poland

Abstract

We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form

z

∈ ϕ

+ Z

0

Z

0

F

(z

)dudυ + Z

0

Z

0

G

(z

)dw

.

Keywords: stochastic inclusions in the plane, set-valued random field, two – parameter stochastic process, weak compactness.

2000 Mathematics Subject Classification: 93E03, 93C30.

1. Introduction

In this paper, we shall consider a stochastic inclusion in a plane of the form:

z _s,t ∈ ϕ _s,t + Z _s

0

Z _t

0

F _u,υ (z _u,υ ) dudυ + Z _s

0

Z _t

D

G _u,υ (z _u,υ ) dw _u,υ . (1)

The results of the paper generalize some problems dealing with stochastic integral equations in a plane of the form:

x _s,t = x + Z _s

0

Z _t

0 a (u, υ, x _u,υ ) dw _u,υ + Z _s

0

Z _t

0 b (u, υ, x _u,υ ) dudυ

(2)

that have been investigated by Tudor [8], Panomarenco [5] and others. The first integral in equation (2) is a stochastic integral in a plane considered with respect to the Wiener-Yeh process. It was defined by Cairoli [1], Panomarenco [5], Tsarenco [7]. Its generalization with respect to two pa- rameters martingale is given by Cairoli and Walsh in [2].

In this paper, we introduce the stochastic integral in a plane for set- valued mappings taking values from space Comp (R ⁿ ) of all nonempty closed subsets of n-dimensional Euclidean space R ⁿ with respect to the Wiener- Yeh process and we establish some of its properties. The space Comp (R ⁿ ) is considered with the Hausdorff metric h defined in the usual way, i.e., h(A, B) = max{h(A, B), h(B, A)}, for A, B ∈ Comp (R ⁿ ) where h (A, B) = sup {dist (a, B) : a ∈ A} and h (B, A) = sup {dist (b, A) : b ∈ B} .

We assume, as given, a complete filtered probability space (Ω, F, (F _s,t ) _s,t≥0 , P ) where a filtration (F _s,t ) _s,t≥0 is assumed to satisfy: F _s,t ⊂ F _u,υ for (s, t) ≤ (u, υ) i.e., for s ≤ u and t ≤ υ. We define F _s,t ¹ = F _s,∞ = ^S _v F _s,v and F _s,t ² = F _∞,t = ^S _u F _u,t . We put R ² ₊ = [0, ∞) × [0, ∞) and β ₊ ² denotes the Borel σ-algebra on R ² ₊ . We consider set-valued two parameter processes (set-valued random fields) (F _s,t ) _s,t≥0 and (G _s,t ) _s,t≥0 that are assumed to be nonanticipative and such that

Z _∞

0

Z _∞

0 kF _s,t k ² dsdt < ∞ and Z _∞

0

Z _∞

0 kG _s,t k ² dsdt < ∞ a.s.,

where kAk := sup{|a| : a ∈ A} for a nonempty set A ⊂ R ⁿ . Next, using the method which has been used by Kisielewicz in [4], we investigate the existence of solutions to stochastic inclusion (1).

2. Basic definitions and notations

Throughout the paper, we shall assume that filtered complete probability space (Ω, F, (F _s,t ) _s,t≥0 , P ) satisfies the following conditions:

(i) if (s, t) ≤ (s ⁰ , t ⁰ ) then F _s,t < F _s

_,t

; (ii) F _0,0 contains all null sets of F;

(iii) for each (s, t), F _s,t = ^T (u,υ)>(s,t) F _u,υ ;

(iv) for each (s, t), F _s,t ¹ and F _s,t ² are conditionally independent given F _s,t .

We write (s, t) ≤ (s ⁰ , t ⁰ ) iff s ≤ s ⁰ and t ≤ t ⁰ . As usual, we shall consider a set R ² ₊ × Ω as a measurable space with the product σ-algebra B ₊ ^{2 N} F.

An n-dimensional two parameter stochastic process z (random field) understood as a function z : R ² ₊ × Ω → R ⁿ with F-measurable sections z _s,t , each s, t ≥ 0, is denoted by (z _s,t ) _s,t≥0 . It is measurable if z is B ₊ ^{2 N} Ω- measurable. The process (z _s,t ) _s,t≥0 is F _s,t -adapted or adapted if z _s,t is F _s,t - measurable for s, t ≥ 0. Every measurable and adapted process is called nonanticipative. In what follows, the Banach space (L ² (R ² ₊ × Ω, β ₊ ^{2 N} F, dsdt ^N P ), k · k ₂ ), where dsdt denotes Lebesgue measure on R ² , will be denoted by L ² _n . Similarly, the Banach spaces (L ² (Ω, F, P, R ⁿ ), k · k) and (L ² (Ω, F _s,t , P, R ⁿ ), k · k) are denoted by L ² (F) and L ² (F _s,t ), respectively.

Let M ² (F _s,t ) denote the family of all (equivalence classes of) n-dimensional nonanticipative processes (f _s,t ) _s,t≥0 such that ^R ₀ ^{∞ R} ₀ ^∞ |f _s,t | dsdt < ∞, a.s.

We shall also consider a subspace L ² (F _s,t ) of M ² (F _s,t ) defined by

L ² (F _s,t ) =

½

(f _s,t ) _s,t≥0 ∈ M ² (F _s,t ) : E Z _∞

0

Z _∞

0 |f _s,t | ² dsdt < ∞

¾ .

It is a closed subset of the Banach space L ² _n . Finally, by M _n (F _s,t ) we de- note a space of all (equivalence classes of) n-dimensional F _s,t -measurable mappings. Throughout, by (w _s,t ) _s,t≥0 we mean two parameter F _s,t -Brownian motion, i.e., a continuous Gausian process such that E(w _s,t ) = 0 and E(w _s,t ·w _s

_,t

) = min(s, s ⁰ )·min(t, t ⁰ ) for every s, s ⁰ , t, t ⁰ . Given g ∈ M ² (F _s,t ), by ( ^R ₀ ^{t R} ₀ ^s g _u,υ dw _u,υ ) _s,t≥0 we denote its stochastic integral with respect to an F _s,t -Brownian motion (w _s,t ) _s,t≥0 . Let us denote by D the family of all n- dimensional F _s,t -adapted continuous processes (z _s,t ) _s,t≥0 such that E sup _s,t≥0 |z _s,t | ² < ∞. The space D is considered as a normed space with the norm | · | _l defined by |x| _l = k sup _s,t≥0 |x _s,t | k _L

for x = (x _s,t ) _s,t ∈ D, where k · k _L

is a norm of L ² (Ω, F, P, R). It can be verified that (D, | · | _l ) is a Banach space. In what follows, we shall deal with upper and lower semi- continuous set-valued mappings. Recall that a set-valued mapping R with nonempty values in a topological space (Y, T _Y ) is said to be upper (lower) semicontinuous [u.s.c.(l.s.c.)] on a topological space (X, T _X ) if R ⁻ (C) :=

{x ∈ X : R(x) ∩ C 6= φ} (resp. R ₋ (C) := {x ∈ X : R(x) ⊂ C}) is a closed

subset of X for every closed set C ⊂ Y. In particular, for R defined on a

metric space (X , d) with values in Comp (R ⁿ ), it is equivalent (see [4]) to

lim _n→∞ h(R(x _n ), R(x)) = 0 (lim _n→∞ h(R(x), R(x _n )) = 0) for every x ∈ X

and every sequence (x _n ) of X converging to x. In what follows, we shall need the following well-known (see [4]) fixed point and continuous selection theorems.

Theorem 1 (Schauder, Tikhonov). Let (X, F _X ) be a locally convex topo- logical Hausdorff space, K a nonempty compact convex subset of X and f a continuonus mapping of K into inself. Then f has fixed point in K.

Theorem 2 (Kakutani, Fan). Let (X, F _X ) be a locally convex topological Hausdorff space, K a nonempty compact convex subset of X and CCl(X) a family of all nonempty closed convex subsets of K. If K : K → CCl(K) is u.s.c. on K, then there exists x ∈ K such that x ∈ K(x).

Theorem 3 (Michael). Let (X, F _X ) be a paracompact space and let K be a set-valued mapping from X to a Banach space (Y, k · k) whose values are closed and convex. Suppose, further K is l.s.c. on X. Then there is a continuous function f : X → Y such that f (x) ∈ K(x), for each x ∈ X.

3. Set-valued stochastic integrals in the plane

Given measure space (X, β, m), a set-valued function R : X → Comp (R ⁿ ) is said to be β-measurable if R ⁻ (C) ∈ β for every closed C ⊂ R ⁿ . For such a multifunction we define integrals subrajectory (see [6]) as a set

S ^p (R) = {g ∈ L ^p (X, β, m, R ⁿ ) : g(x) ∈ R(x) a.e. }.

It is clear that for nonemptiness of S ^p (R) we have to assume more than β-measurability of R. In what follows, we shall assume that β-measurable set-valued function R : X → Comp (R ⁿ ) is p-integrably bounded, p ≥ 1, i.e., that a real-valued mapping X 3 x → kR(x)k ∈ R ₊ belongs to L ^p (X, β, m, R ₊ ) where kAk := sup{|a| : a ∈ A} for A ∈ Comp (R ⁿ ). It can be verified that a β-measurable set-valued mapping R : X → Comp (R ⁿ ) is p-integrably bounded p ≥ 1, if and only if S ^p (R) is nonempty and bounded in L ^p (X, β, m, R ⁿ ) (see [4]). Finally, it is easy to see that S ^p (R) is decompos- able, i.e., such that 1| _A f ₁ + 1| _X\A f ₂ ∈ S ^p (R) for A ∈ β and f ₁ , f ₂ ∈ S ^p (R).

We have the following general result dealing with properties of subtrajectory integrals (see [4], [5]).

Proposition 4. Let R : X → Comp (R ⁿ ) be β-measurable and p-integrably

bounded, p ≥ 1. The set S ^p (R) is a nonempty bounded and closed subset

of L ^p (X, β, m, R ⁿ ). Moreover, if R takes on convex values, then S ^p (R) is convex and weakly compact in L ^p (X, β, m, R ⁿ ).

Let G = (G _s,t ) _s,t≥0 be a set-valued two parameter stochastic process with values in Comp(R ⁿ ), i.e., a family of F-measurable set-valued mappings G _s,t : Ω → Comp(R ⁿ ), s, t ≥ 0. We call G measurable if it is β ₊ ^{2 N} F- measurable. Similarly, G is said to be F _s,t -adapted or adapted if G _s,t is F _s,t - measurable for each s, t ≥ 0. A measurable and adapted set-valued two pa- rameter stochastic process is called nonanticipative. Denote by M ² _s−υ (F _s,t ) a family of all nonanticipative set-valued processes G = (G _s,t ) _s,t≥0 such that R _∞

0

R _∞

0 kG _s,t k ² dsdt < ∞, a.s. Immediately from the Kuratowski and Ryll- Nardzewski measurable selection theorem (see [4]) it follows that for every F, G ∈ M ² _s−υ (F _s,t ) their subtrajectory integrals

S ² (F ) = f ∈ {M ² (F _s,t : f _s,t ∈ (ω) ∈ F _s,t (ω), dsdt × P − a.e. } and

S ² (G) = {g ∈ M ² (F _s,t ) : g _s,t (ω) ∈ G _s,t (ω), dsdt × P − a.e. },

are nonempty. Indeed, let ^P = {Z ∈ B ² ₊ ^N F : Z _s,t ∈ F _s,t , each s, t ≥ 0}, where Z _s,t denotes a section of Z determined by s, t ≥ 0. It is a σ-algebra on R ² ₊ × Ω and function f : R ² ₊ × Ω → R ⁿ (a multifunction F : R ² ₊ × Ω → Comp (R ⁿ )) is nonanticipative if and only if it is ^P -measurable. Therefore, by the Kuratowski and Ryll-Nardzewski measurable selection theorem, every nonanticipative set-valued function admits a nonanticipative selector. It is clear that for F ∈ M ² _s−υ (F _s,t ) such selectors belong to M ² (F _s,t ). Finally, denote

L ² _s−υ (F _s,t ) =

½

G ∈ M _s−υ (F _s,t : E Z _∞

0

Z _∞

0 kG _s,t k ² dsdt < ∞

¾ . Given set-valued two parameter processes

F = (F _s,t ) _s,t≥0 ∈ M ² _s−υ (F _s,t ) and G = (G _s,t ) _s,t≥0 ∈ M ² _s−υ (F _s,t ) by their stochastic integrals we mean families ( ^R ₀ ^{s R} ₀ ^t F _u,υ dudυ) _s,t≥0 and ( ^R ₀ ^{s R} ₀ ^t G _u,υ dw _u,υ ) _s,t≥0 of subsets of M _n (F _s,t ), defined by

Z _s

0

Z _t

0 F _u,υ dudυ :=

½Z _s

0

Z _t

0 f _u,υ dudυ : f ∈ S ² (F )

¾

and _Z

s 0

Z _t

0 G _u,υ dw _u,υ :=

½Z _s

0

Z _t

0 g _u,υ dw _u,w : g ∈ S ² (G)

¾ .

Immediately, from the above definitions the following simple results follow.

Theorem 5. Let F, G ∈ M ² _s−υ (F _s,t ). Then

(i) ^R ₀ ^{s R} ₀ ^t F _u,υ dudυ and ^R ₀ ^{s R} ₀ ^t G _u,υ dw _u,υ are nonempty subsets of M _n (F _s,t ), each s, t ≥ 0. They are convex if F and G take on convex values.

(ii) If G ∈ L ² _s−υ (F _s,t ), then ^R ₀ ^{s R} ₀ ^t G _u,υ dw _u,υ is a nonempty subset of L ² (F _s,t ), each s ≥ 0, t ≥ 0,

(iii) If F, G ∈ L _R ² _s−υ (F _s,t ) take on convex values then ^R ₀ ^{s R} ₀ ^t F _u,υ dudυ and

0 s

R _t

0 G _u,υ dw _u,υ are nonempty convex and weakly compact subsets of L ² (F _s,t ), each s ≥ 0, t ≥ 0.

P roof. It is clear that (i) follows immediately from the definitions of set- valued stochastic integrals. To verify (ii) let us obserwe that the operator I _s,t defined by the formula I _s,t g = ^R ₀ ^{s R} ₀ ^t g _u,υ dw _u,υ is a linear continuous mapping from L ² (F _s,t ) into L ² (F _s,t ). By the properties of stochastic integrals in the plane (see [6]) one has E|I _s,t | ² = kgk ² ₂ , each s, t ≥ 0 and g ∈ L ² (F _s,t ), where k · k ₂ denotes the norm of L ² (F _s,t ). Therefore, I _s,t maps closed subsets of L ² (F _s,t ) into closed subsets of L ² (F _s,t ). Thus (ii) is satisfied. To verify (iii), it sufficies only to observe that for every s, t ≥ 0, all sets S ² (F ) and S ² (G) are convex and weakly compact in L ² (F _s,t ). Hence, J _s,t (S ² (F )) and I _s,t (S ² (G)) (where J _s,t , is defined by J _s,t f = ^R ₀ ^{s R} ₀ ^t f _s,t dudυ) are convex and weakly compact subsets of L ² (F _s,t ) because J _s,t and I _s,t are linear and continuous for fixed s, t ≥ 0. Then (iii) also holds.

4. Stochastic inclusions in the plane

Let F = {(F _s,t (z)) _s,t≥0 : z ∈ R ⁿ } and G = {(G _s,t (z)) _s,t≥0 : z ∈ R ⁿ }. Assume F and G are such that (F _s,t (z)) _s,t≥0 ∈ M ² _s−υ (F _s,t ) and (G _s,t (z)) _s,t≥0 ∈ M ² _s−υ (F _s,t ) for each z ∈ R ⁿ . By a stochastic inclusion in the plane we mean the relation

z _s,t ∈ z _0,0 + Z _s

0

Z _t

0 F _u,υ (x _u,υ ) dudυ + Z _s

0

Z _t

0 G _u,υ (z _u,υ ) dw _u,υ .

(3)

Every two parameter stochastic process (z _s,t ) _s,t≥0 ∈ D such that F _s,t (z _s,t ) ∈ M ² _s−υ (F _s,t ) and G _s,t (z _s,t ) ∈ M ² _s−υ (F _s,t ) satisfying a.s. the relation (3) is said to be a global solution to (3). Suppose F and G satisfy the following conditions (C ₁ )

(i) F = {(F _s,t (z)) _s,t≥0 : z ∈ R ⁿ } and G = {(G _s,t (z)) _s,t≥0 : z ∈ R ⁿ } are such that mappings R ² ₊ × Ω × R ⁿ 3 (s, t, w, z) → F _s,t (z) ∈ Cl(R ⁿ ) and R ² ₊ × Ω × R ⁿ 3 (s, t, w, z) → G _s,t (z) ∈ Cl(R ⁿ ) are β ₊ ^{2 N} Ω ^N β ⁿ measurable.

(ii) (F _s,t (z) _s,t≥0 ) and (G _s,t (z)) _s,t≥0 are uniformly square-integrably bounded, i.e.,

sup

z∈R

kF _s,t (z)k ∈ L ² ₁ and sup

z∈R

kG _s,t (z)k ∈ L ² ₁ .

Now define a linear continuous mapping Φ on L ² _n × L ² _n taking Φ(f, g) = (I _s,t f + J _s,t g) _s,t≥0 for (f, g) ∈ L ² _n × L ² _n . It is clear that Φ maps L ² _n × L ² _n into D. Let ϕ = ϕ _s,t ∈ D be given. For F and G satisfying (C ₁ ) define a set-valued mapping K on D by setting

K(z) = ϕ + Φ(S ² F _s,t (z)) _s,t≥0 × S ² (G _s,t (z)) _s,t≥0 for z = (z _s,t ) _s,t≥0 ∈ D.

Let F and G satisfy conditions (C ₁ ) and the following condition (C ₂ ) : set-valued functions

D 3 z → (F _s,t (z)) _s,t≥0 (ω) ⊂ R ⁿ and D 3 z → (G _s,t (z)) _s,t≥0 (ω) ⊂ R ⁿ are w-w.s.u.s.c. on D, i.e., for every z ∈ D and every sequence (z _n ) of (D, k · k _l ) converging weakly to z, one has

h Z Z

A

Z ³

F _s,t ^³ z ⁽ⁿ⁾ _s,t ^´´ dsdtdP, Z Z

A

Z

(G _s,t (z _s,t )) dsdtdP ) → 0 and

h Z Z

A

Z ³

G _s,t ^³ z _s,t ^{(n) ´´} dsdtdP, Z Z

A

Z

(G _s,t (z _s,t )) dsdtdP → 0, for every A ∈ β ₊ ^{2 N} Ω.

Lemma 6. Assume F and G take on convex values and satisfy (C ₁ ) and

(C ₂ ). Then a set-valued mapping K is u.s.c. as a multifunction defined on

a locally convex topological Hausdorff space (D, σ(D, D ^∗ )) with nonempty

values in (D, σ(D, D ^∗ )).

P roof. Let C be a nonempty weakly closed subset of D and select a se- quence (z ⁿ ) of K ⁻ (C) weakly converging to z ∈ D. There is a sequence (y ⁿ ) of C such that y ⁿ ∈ K(z _n ) for n = 1, 2, ... . By the uniform square- integrable boudedness of F and G, there is a convex weakly compact subset B ⊂ L ² _n × L ² _n such that K(z _n ) ⊂ Φ(B). Therefore, y ⁿ ∈ Φ(B) for n = 1, 2, ...

which, by the weak compactness of Φ(B) implies the existence of a subse- quence (y ^k ) of (y ⁿ ) weakly converging to y ∈ Φ(B). We have y ^k ∈ K(z ^k ) for k = 1, 2, ... . Let (f ^k , g ^k ) ∈ S ² (F _s,t (z _s,t ^k )) × S ² (G _s,t (z ^k _s,t )) be such that Φ(f ^k , g ^k ) = y ^k , for each k = 1, 2, ... . We have of course (f ^k , g ^k ) ∈ B. There- fore, there is a subsequence, say again (f ^k , g ^k ) of (f ^k , g ^k ) weakly converging in L ² _n × L ² _n to (f, g) ∈ B. Now, for every A ∈ β ² ₊ ^N Ω one obtains

dist



 Z Z

A

Z

f _s,t dsdtdP, Z Z

A

Z

F _s,t (z) dsdtdP



 ≤

¯ ¯

¯ ¯

¯ ¯ Z Z

A

Z ³

f _s,t − f _s,t ^{k ´} dsdtdP

¯ ¯

¯ ¯

¯ ¯

+ dist



 Z Z

A

Z

f _s,t ^k dsdtdP, Z Z

A

Z

F _s,t ^³ z ^{k ´} dsdtdP





+ h



 Z Z

A

Z

F _s,t (z ^k ) dsdtdP, Z Z

A

Z

F _s,t (z) dsdtdP



 .

Therefore, f ∈ S ² (F _s,t (z _s,t )) _s,t≥0 . Quite similarly, we also get g ∈ S ² (G _s,t (z _s,t ) _s,t≥0 ). Thus, ϕ + Φ(f, g) ∈ K(z), which implies y ∈ K(z). On the other hand, we also have y ∈ C, because C is weakly closed. Therefore, z ∈ K ⁻ (C). Now the result follows immediately from Eberlein and ˆ Smulian’s Theorem.

Theorem 7. If F and G take on convex values and satisfy (C ₁ ) and (C ₂ ), then there is z ∈ D such that

z _s,t ∈ ϕ _s,t + Z _s

0

Z _t

0 F _u,υ (z _u,υ ) dudυ + Z _s

0

Z _t

0 G _u,υ (z _u,υ ) dw _u,υ . P roof. Let

{B = (f, g) ∈ L ² _n × L ² _n : |f _s,t (ω)| ≤ kF _s,t (ω)k, |g _s,t (ω)| ≤ kG _s,t (ω)k}

and put K = ϕ + Φ(B). It is clear that K is a nonempty convex weakly

compact subset of D such that K(z) ⊂ K for z ∈ D. By (iii) of Theorem 5

K(z) is a convex and weakly compact subset of D, for each z ∈ D. By Lemma 6 K is u.s.c. on a locally convex topological Hausdorff space (D, σ(D, D ^∗ )). Therefore, by the Kakutani and Fan fixed point theorem there exists z ∈ K such that z ∈ K(z), i.e.,

z _s,t ∈ ϕ _s,t + Z _s

0

Z _t

0 F _u,υ (z _u,υ ) dudυ + Z _s

0

Z _t

0 G _u,υ (z _u,υ ) dw _u,υ .

Assume now that F and G satisfy condtions (C ₁ ) and the following condition (C ₃ ) F and G are such that set-valued functions:

D 3 z → (F _s,t (z) _s,t≥0 (ω) ⊂ R ⁿ ) and D 3 z → (G _s,t (z) _s,t≥0 (ω) ⊂ R ⁿ ) are s.-w.s.l.s.c. on D, i.e., for every z ∈ D and every sequence (z _n ) of (D, | | _l ) converging weakly to z one has

h ^h (F _s,t (z)) _s,t≥0 (w), (F _s,t (z _n )) _s,t≥0 (ω) ⁱ → 0 and

h ^h (G _s,t (z)) _s,t≥0 (w), (G _s,t (z _n )) _s,t≥0 (ω) ⁱ → 0 a.e.

Lemma 8. Assume F and G take on convex values and satisfy (C ₁ ) and (C ₃ ). Then a set-valued mapping K is l.s.c. as a multifunction defined on a locally convex topological Hausdorff space (D, σ(D, D ^∗ )).

P roof. Let C be a nonempty weakly closed subset of D and (z ⁽ⁿ⁾ ) a se- quence of R ₋ (C) weakly converging to z ∈ D. Select arbitrary u ∈ K(z) and suppose (f, g) ∈ S ² (F _s,t (z) _s,t≥0 ) × S ² (G _s,t (z) _s,t≥0 ) is such that u = Φ(f, g).

Let (f ⁽ⁿ⁾ , g ⁽ⁿ⁾ ) ∈ S ² (F _s,t (z ⁽ⁿ⁾ ) _s,t≥0 ) × S ² (G _s,t (z ⁿ ) _s,t≥0 ) be such that

¯ ¯

¯ f _s,t (ω) − f _s,t ⁽ⁿ⁾ (ω)

¯ ¯

¯ = dist ^³ f _s,t (ω), ^³ F _s,t ^³ z ^{(n) ´´} (ω) ^´

and _¯

¯ ¯ g _s,t (ω) − g _s,t ⁽ⁿ⁾ (ω) ^{¯ ¯} ¯ = dist ^³ g _s,t (ω), ^³ G _s,t (z ⁽ⁿ⁾ ) ^´ (ω) ^´

on R ² ₊ × Ω, for each n = 1, 2, ... . By virtue of (C ₃ ) one gets |f _s,t (ω) −

f _s,t ⁽ⁿ⁾ (ω)| → 0 and |g _s,t (ω) − g ⁽ⁿ⁾ _s,t (ω)| → 0 a.e., as n → ∞. Hence by (C ₁ )

we can see that a sequence (u ⁽ⁿ⁾ ), defined by u ⁽ⁿ⁾ = Φ(f ⁽ⁿ⁾ , g ⁽ⁿ⁾ ) weakly

converges to u. But u ⁽ⁿ⁾ ∈ K(z ⁽ⁿ⁾ ) ⊂ C for n = 1, 2, ... and C is weakly

closed. Then u ∈ C, which implies K(z) ⊂ C. Thus z ∈ K ₋ (C).

Theorem 9. If F and G take on convex values and satisfy (C ₁ ) and (C ₃ ) then stochastic integral inclusion (1) admits a solution.

P roof. Let

B = {(f, g) ∈ L ² _n × L ² _n : |f _s,t (ω)| ≤ kF _s,t (ω)k, |g _s,t (ω)| ≤ kG _s,t (ω)k}

and put K = ϕ + Φ(B). It is clear that K is a nonempty convex weakly compact subset of D such that K(z) ⊂ K for z ∈ D. By virtue of Lemma 8, K is l.s.c. as a set-valued mapping from a paracompact space K considered with its relative topology induced by a weak topology σ(D, D ^∗ ) on D into a Banach space (D, k · k _l ). By (iii) of Theorem 5, K(z) is a closed and convex subset of D, for each z ∈ K. Therefore, by Michael’s theorem, there is a continuous selection k : K → D for K. But K(K) ⊂ K. Then K maps K into itself and is continuous with respect to the relative topology on K, defined above. Therefore, by the Schauder and Tikhonov fixed point theorem, there is z ∈ K such that z = k(z) ∈ K(z), i.e.,

z _s,t ∈ ϕ _s,t + Z _s

0

Z _t

0 F _u,υ (z _u,υ ) dudυ + Z _s

0

Z _t

0 G _u,υ (z _u,υ ) dw _u,υ a.s.

References

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Sc. Paris 274 (A) (1972), 1739–1742.

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Stoch. Anal. 6 (3) (1993), 217–236.

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Received 10 October 2001

Revised 18 November 2001