PROPERTIES OF SOLUTION SET OF STOCHASTIC INCLUSIONS

The properties of the solution set of stochastic inclusions xt−xs∈clL2(∫stFτ(xτ)dτ


INTRODUCTION
There is a large number of papers (see for example [11, [4] and [5]) dealing with the existence of optimal controls of stochastic dynamical systems described by integral stochastic equations.Such problems can be described (see [10]) by stochastic inclusions (SI(F, G,H))of the form x x, e ClL F,.(x.)dr + G(z)dw + H,(x)V (d,, dz)   8 8 8 n where the stochastic integrals are defined by Aumann's procedure (see [7], [9]).
The results of the paper are concerned with properties of the set of all solutions to SI(F,G,H).To begin with, we recall the basic definitions dealing with set-valued stochastic integrals and stochastic inclusions presented in [10].We assume, as given, a complete filtered probability space (f,,(tt)t>o,P), where a family (t)t>0, of a-algebras 5t C is assumed to be increasing: .C t if s _< t.We set N+ [0, c), and %+ will denote the Borel a-algebra on 1Received: April, 1993.Revised: July, 1993.R+.We consider set-valued stochastic processes (F,),_>o,(,),>o and (t.), >o..e", aking on values from he space Comp(l") of all nonempy compacg subsegs of n-dimensional Euclidean space N".They are assumed go be predictable and such ha E f I I , I I dt < oo, p >_ 1, E f I I g, I I dt < oo d o o i i I I ,,, I I 2dtq(dz) < oo, where q is a meazure on he Borel a-algebra %' of oRn d I I A I1" = p{l I' e A}, A e Comp(").The space Comp(N") is considered with the Hausdorff metric h defined in the usual way, i.e., h(A, B) = maz{ (A, B), (B, A)}, for A, B Comp(R"), where (A, B)   = {dist(a, B): a e A} and (B, A) = {dist(b, A): b e B}.Although the classical theory of stochastic integrals (see [3], IS], [12]) usually deals with measurable and t-adapted processes, it can be finally reduced (see [4], pp.60-62) to predictable ones.

BASIC DEFINITIONS AND NOTATIONS
Throughout the paper we shall assume that a filtered complete probability space (f, q, (t)t > o, P) satisfies the following usual hypotheses" (i)qo contains all the P-null sets of , (ii) -V t> ot and (iii) t = f'l , for all u>t t, 0 _< t < c.As usual, we consider a set N+ x f as a measurable space with the producg a-algebra N+ (R) .Moreover, we introduce on N+ x f he predictable -algebra generated by a semiring % of all predictable rectangles in [+ x f of the form {0} xAo and (s,t]xA,, where A o o and A for s < t in N+.
Similarly, besides the usual product a-algebra on N+ x f x N", we also introduce ghe predicgable o'-algebra ' generaged by a semiring %" of all segs of ghe form {0}xA 0xD and (s,t]xAxD, with Aoabo, A,, for s<t in N+ and D %, where N consists of all Borel sets D C N" such that their closure does not contain the point 0.
An n-dimensional stochastic process x, understood as a function x:+ x g/N" with Zb-measurable sections xt, each t > 0, is denoted by (xt)t >_o.
It is measurable (predictable) if z is %+ (R) (, resp.)-measurable.The process (Xt)t > 0 is t-adapted if z is qt-measurable for t > 0. It is clear (see thag every predicgable process is measurable and :t-adapged.In whag follows ghe Banach space LP( + f, , dt P, "), p >_ 1, with the norm I I I I zg defined in ghe usual way, will be denoted by .Similarly, the Banach spaces L'(12, Jt, P,") and L(f, ff, P,") with the usual norm II-II g e denoted by L,(t) and L(), respectively.Throughout the paper, by (w,)t>o, we mean a one-dimensional Brownian motion starting at 0, i.e., such that P(wo = 0)= 1.By u(t,A) we denote a t-Poissoa measure on N+ x N", and then define a t-ceatered Poisson measure P (t, A), t > 0, A %", by taking V (t,A)= (t,A)-tq(A), t >_ O, A e where q is a measure on " such that Eu(t, B)= tq(B) and q(B) < o0 for B e %.
Let us denote by D the family of all n-dimensional t-adapted cdlg pro- Recall that an n-dimensional stochasgic process is said to be a cgdlgg process if it has almost all sample paths right continuous with finite left limits.The space D is considered as a normed space with the norm II'lle defined by I I I I a = I I p,_> o I, I I o = (,),_> o E O.It can be verified that (O, I1" I I )is a Banach space.
Given 0 a < < d (xt)t > o E D let x"'a= (xT'a), > o be defined by x ' = x and x ' = x for 0 t a d t fl, respectively, d z ' = x for t .It is cle that D': = {z'o:z e D} is a line subspace of D, closed in he I1" I I -om opology.hen (D', I1" I I )i so gc PinMly, usual, by (D,D') we shall denote a weak gopology on D.
In what follows we shall deal with upper and lower semicontinuous set- valued mppings.Recall that a set-valued mapping % with nonempty values in a topological space (Y,y)is said to be upper (lower) semicontinuous [u.s.c.(1.s.c.)] on a topological space (X,x)if -(C): (%_(0): = {x e X:%(x)C C}) is a closed subset of X for every closed set C C Y. In particular, for % defined on a metric space (%,d) with values in Comp(R"), it is equivalent (see [9]) to lih (%(x,), %(x)) = 0 (li.ooK (%(x), %(x,)) = O) for every converging to x. If, moreover, % takes convex values then it is equivalent to upper (lower) semicontinuity of a real-vlued function s(p, %(. ))on " for every p R", where s(., A) denotes a support function of a set A Comp(").In what follows, we shall need the follow well-known (see [9]) fixed point and continuous selection theorems.
Theorem (Schauder, Tikhonov)" Let (X, ffx) be a locally convex topological Hausdorff space, % a nonempty compact convex subset of X and f a continuous mapping of % into itself.Then f has a fixed point in %.
Theorem (Covitz, Nadler)" Let (%,d) be a complete metric space and ):---,C/() a set-valued contraction mapping, i.e., such that H((x), (y)) < d(x, y) for x, y e % with E [0,1), where H is the Hausdorff metric induced by the metric d on the space Cl(%) of all nonempty closed bounded subsets of %.Then there exists x E such that x E %(x).
Theorem (Kakutani, Fan)" Let (X,x) be a locally convex topological Hausdorff space, % a nonempty compact convex subset of X and COl(%) a family of all nonempty closed convex subsets of %.If : %---}CC1(%) is u.s.c, on % then there exists x E % such that x E Theorem (Michael)" Let (X, ffx) be a paracompact space and let % be a set-valued mapping from X to a Banach space (Y, I1" II)   whose values are closed and convez.Suppose, further % is l.s.c, on X.Then there is a continuous function f: XY such that f(x) %(x), for each x X.
Similarly, is said to be 9t-adapted if t is 9t-measurable for each t >_ 0. It is clear that every predictable set-valued stochastic process is measurable and 9tadapted.It follows from the Kuratowski and Ryll-Nardzewski measurable selection theorem (see [9]) that every measurable (predictable) set-valued process with nonempty compact values possesses a measurable (predictable) selector.
The following properties of Proposition 1: Let F ./fi ( P ) p >_ l .A o ( P ) and Then t and t% are closed subsets of L(t) for each t >_ O.
If, moreover, F, and , take on convex values then tF, t and t% are convex and weakly compact in L(t) and L(t), respectively, for each t > O.
Corollary 4: For every F,G and H taking on convex values and satisfying (Jt), one has 3( and Let S(F,G,H) and S'a(F,G,H) denote the set of all fixed points of :E and N"O, respectively.It will be showa below that S''O(F, G, H) C. D'.Immediagely from Proposition 2 (see [101) ghe following resulg follows.
B B (A3) F, C and H are such that set-valued functions D x(F o mx),(w) C a", D x---, (G mx)(w) C N" and D x(H mx),,,(w) C N" are s.-w.s.l.s.c.
on D, i.e., for eve x e D and eve sequence (x,) of (D, 11.I I h(H o rex, H o my)II S E: f m,, ,-Y, dtq(dz) for x, y D.
() There re k, e ( +) d m e ( + x ) sch that h(F,()(w), It is clear that the upper (lower) semicontinuity of F, G and H does not imply their wek (strong) weak sequential upper (lower) semicontinuity presented above.We shall show that in some special cases, i.e., for concave (convex, rasp.), set-valued mappings such implication holds true.Recall a set- valued mapping %, defined on a locally convex topological space (X, CJ'x) with values in a normed space is said to be concave (convex)if .%(cx+ fizz)C (x) + Z() ((a)+ Zu() c ( + Z)), or ev , e X nd a,/3 e [0,1] satisfying c +/3 1.

Lamina 1:
Suppose F,G and H satisfy (A1) with p-1, take on convex values and are concave (convex} with respect to x n.If moreover F, G and H are u.s.c.(.s.c.) with respect to x n then they are w.-w.s.u.s.c.(s.- w.s.l.s.c.).
Proof: Let x D be fixed and let (x") be a sequence of D weakly converging to x. Denote K,(t, w, y): = s(p, Ft(yt)(w)) for p e ", y e D, t >_ 0 and w f.We shall show that for every A and every p " one has f f Kp(t,w,z)dtdP <_ liu_nf / f Ko(t,w,x)dtdP, A A which is equivalent to the weak-weak sequential upper semicontinuity of F at x D in the sense defined in (A).Similarly, the weak-weak sequential upper Finally, we ge Kp( z)-< limin f[limin f N] = 5.PR.OPEKTIES OF SOLUTION SET We shall prove here the existence theorems for SI(F,G,H).We show first that conditions (A) and anyone of conditions (A)-(A)or (A)imply the existence of fixed points for the set-valued mappings and ' defined above.
Hence, by Propositions 4 and 5, the existence theorems for SI(F,G,H) will follow.We begin with the following lemmas.
Lemma 2: Assume F,G and H take on convex values, satisfy with p = 2 and (Az).Then a se-valued mapping 3g is u.s.c, as a mulifunction defined on a locally convex topological Hausdorff space (D, cr(D,D*)) with otu ,t (D, o(D, D*)). Proof: Let C be a nonempty weakly closed subset of D and select a sequence (x")of %-(C) weakly converging to x D. There is a sequence (y")of C such that y" %(x") for n = 1,2, By the uniform square-integrable boundedness of F,G and H, there is a convex weakly compact subset % C x L x <W such that %(x")C ().Therefore, y"e (%), for n = 1,2,... which, by the weak compactness of if(%), implies the existence of a subsequence, say for simplicity (y), of (y")weakly converging to y if(%).We have e () o = 1, e, t (p, a, ) e (F o m ) (a o m ) ](H o ) be such that ?(fk, gk, h)=yk, for each k-1,2, We have of course (f,g,h) %.Therefore, there is a subsequence, say again {(f,g,h)} of {(f,gk, hk)} weakly converging in x x 'IV to (f,g,h)e .Now, for every A 9 one obtains < i i [ f t _ .A ft]dtdP-t" d T .s : l ; ( i f i i f(w)-f(w) = dist(f(w), (F o my)t(w)) g(w)-"(w) = dist(g(w), (G o my),(w)) and h' ' h' z(w), (H o my)t,z(w)) on R + x a and R + x a x an respecgively.Now, by (4) ig follows oSuPt > o suPt < < + xt xs = O.