FUZZY STOCHASTIC INTEGRAL EQUATIONS

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1 Dynamic Systems and Applications FUZZY STOCHASTIC INTEGRAL EQUATIONS MAREK T. MALINOWSKI AND MARIUSZ MICHTA Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra, Szafrana 4a, Zielona Góra, Poland, ABSTRACT. In this paper we propose a new approach to fuzzy stochastic integrals of Itô and Aumann type. Then a fuzzy equation with fuzzy stochastic integrals is investigated. The existence and uniqueness of solution is proven. Some typical properties of the solution are also obtained. Similar results to set-valued stochastic integral equations are stated. Keywords and phrases. Fuzzy set, fuzzy stochastic process, fuzzy stochastic integral, fuzzy stochastic differential equation, fuzzy stochastic integral equation, set-valued stochastic integral equation AMS MOS Subject Classification. 6H5, 6H2, 37H1, 3E72 1. INTRODUCTION The theory of fuzzy differential equations has focused much attention in the last decades since it provides good models for dynamic systems under uncertainty. In [18] this theory has been started with a concept of H-differentiability for fuzzy mappings introduced in [42]. Currently the literature on this topic is very rich. For a significant collection of the results and further references we refer the reader to the monographs [9, 27] and to the research articles e.g. [1, 2, 7, 8, 15, 26, 28, 37, 38, 39, 4]. Recently some results have been published concerning random fuzzy differential equations see [11, 12, 31]. The random approach can be adequate in modelling of the dynamics of real phenomena which are subjected to two kinds of uncertainty: randomness and fuzziness, simultaneously. Here fuzzy random variables and fuzzy stochastic processes play a crucial role. The next natural step in modelling of dynamic systems under uncertainty should be the theory of fuzzy stochastic differential equations understood in their integral form. Here, a main problem is the notion of stochastic fuzzy Itô integral. A first background for such a kind of research has been made in [19, 17, 3]. Although these papers differ from each other in the considered settings, the main idea is always the same: to define a stochastic set-valued Itô integral as a measurable set-valued mapping and then, by using the Stacking Theorem, to introduce a concept of stochastic Received March 1, $15. c Dynamic Publishers, Inc.

2 474 M. T. MALINOWSKI AND M. MICHTA fuzzy integral as a fuzzy stochastic process. The authors of [19, 3] emphasize their motivations to such studies as the beginning of fundations to the theory of fuzzy stochastic differential equations and inclusions. As far as we know there are two papers concerning this new area, i.e. [2] and [41]. However the approaches presented there are different. In [2] all the considerations are made in the setup of fuzzy sets space of a real line, and the main result on the existence and uniqueness of the solution is obtained under very particular conditions imposed on the structure of integrated fuzzy stochastic processes such that a maximal inequality for fuzzy stochastic Itô integrals holds. Unfortunately the paper [2] contains the gaps. In view of [44] we find out that the intersection property of a set-valued Itô integral defined as a measurable set-valued mapping may not hold true in general. Thus a definition of fuzzy stochastic integral, which is used in [2], seems to be incorrect. On the other hand, in [41] a proposed approach does not contain a notion of a fuzzy stochastic Itô integral. The method presented there is based on sets of appropriately chosen selections. In [32] we proposed a third approach to stochastic fuzzy differential equations. We gave a result of existence and uniqueness of the solution to stochastic fuzzy differential equation where the diffusion term appropriate fuzzy stochastic Itô integral was of some special form, i.e. it was the embedding of real d-dimensional Itô integral into fuzzy numbers space. In this paper we propose completely different approach to the notions of fuzzy stochastic integrals. We employ here the notion of set-valued stochastic integral which was widely used in the theory of stochastic inclusions and their applications see e.g. [3, 4, 5, 6, 22, 23, 24, 25, 33, 34, 35] and references therein. As distinct from the approaches in [19, 29, 3, 32, 41] we treat the fuzzy stochastic integrals as the fuzzy sets of the space of square-integrable random vectors. Then we consider the fuzzy integral equations in which these fuzzy stochastic integrals appear. The paper is organized as follows. In Section 2 we recall some facts from the set-valued stochastic analysis. We give also the definitions and some properties of set-valued trajectory stochastic integrals of Itô and Aumann type. The Section 3 is started with a short résumé about fuzzy sets and fuzzy stochastic processes. Next the results of preceding section are used to establish the notions of fuzzy trajectory stochastic integrals of Itô and Aumann type. Some properties of these fuzzy stochastic integrals are also stated. Finally a fuzzy stochastic integral equation, with fuzzy trajectory stochastic Itô integral and fuzzy trajectory stochastic Aumann integral, is investigated. We present the results on existence and uniqueness of the solution as well as on boundedness of the solution, on continuous dependence on initial conditions and on the stability of the solution. In Section 4 we formulate the parallel results which are established for set-valued stochastic integral equations.

3 FUZZY STOCHASTIC INTEGRAL EQUATIONS PRELIMINARIES Let X be a separable Banach space, K b X the family of all nonempty closed and bounded subsets of X. Similarly by K b cx we denote the family of all nonempty closed, bounded and convex subsets of X. The Hausdorff metric H X defined by { H X A, B = max sup a A } dist X a, B, sup dist X b, A, b B in K b X is where dist X a, B = inf a b X and X denotes a norm in X. b B It is known that K b X, H X is a complete metric space, Kc b X is its closed subspace. For nonempty subsets A 1, A 2, B 1, B 2 of X it holds see [14] H X A 1 + A 2, B 1 + B 2 H X A 1, B 1 + H X A 2, B 2, where A 1 + A 2 denotes the Minkowski sum of A 1 and A 2. Let U, U, µ be a measure space. Recall that a set-valued mapping F : U K b X is said to be measurable if it satisfies: {u U : Fu C } U for every closed set C X. A measurable multifunction F is said to be L p -integrably bounded p 1, if there exists h L p U, U, µ; R + such that the inequality F h holds µ-a.e., where A = H X A, {} = sup a X a A for A K b X, and R + = [,. Consequently, it is known see [13] that F is L p -integrably bounded if and only if F L p U, U, µ; R +. Let M be a set of U-measurable mappings f : U X. The set M is said to be decomposable if for every f 1, f 2 M and every A U it holds f 1 1 A + f 2 1 U\A M. Denote I = [, T], where T <. Let Ω, A, {A t }, P be a complete filtered probability space satisfying the usual hypotheses, i.e. {A t } is an increasing and right continuous family of sub-σ-algebras of A and A contains all P-null sets. Let {Bt} be an {A t }-adapted Wiener process. We put U = I Ω, U = N, where N denotes the σ-algebra of the nonanticipating elements in I Ω, i.e. N = {A β I A : A t A t for every t I}, where β I is the Borel σ-algebra of subsets of I and A t = {ω : t, ω A} for t I. Finally we set µ = λ P as a measure, where λ is the Lebesgue measure on I, β I. A d-dimensional stochastic process f : I Ω R d is called nonanticipating if f is N-measurable. Clearly N β I A. It is known that f is N-measurable if and only if f is β I A-measurable and {A t }-adapted. Consider the space L 2 N λ P := L2 I Ω, N, λ P; R d.

4 476 M. T. MALINOWSKI AND M. MICHTA Then for every f L 2 N λ P and, t I, < t the Itô stochastic integral fsdbs exists and one has fsdbs L2 Ω, A t, P; R d L 2 Ω, A, P; R d. Moreover E fsdbs 2 t = E fs 2 ds = [,t] Ω f 2 ds dp, where denotes usual Euclidean norm in R d see [16] for details. Let F : I Ω K b R d be a set-valued stochastic process, i.e. a family {Ft} of A-measurable set-valued mappings Ft: Ω K b R d, t I. We call F nonanticipating if it is N-measurable. Let us define the set S 2 NF, λ P := {f L 2 Nλ P : f F, λ P-a.e.}. If F is L 2 N λ P-integrably bounded, then by Kuratowski and Ryll-Nardzewski Selection Theorem see e.g. [21] it follows that SN 2 F, λ P. Hence for every, t I, < t we can define the set-valued trajectory Itô stochastic integral J t F := { fsdbs : f S 2 N F, λ P }. Remark 2.1. By the above definition we have J t F L2 Ω, A, P; R d. In the rest of the paper, for the sake of convenience, we will write L 2 instead of L 2 Ω, A, P; R d. Proposition 2.2. Let F : I Ω K b R d be a nonanticipating and L 2 N λ P- integrably bounded set-valued stochastic process. Then i SN 2 F, λ P is nonempty, bounded, closed, weakly compact and decomposable subset of L 2 N λ P, ii J tf is nonempty, bounded, closed and weakly compact subset of L2 Ω, A t, P; R d for every, t I, < t. Proof. The proof of part i follows immediately from the assumptions imposed on F. The decomposability of SN 2 F, λ P follows by Theorem 3.1 in [13]. For the proof of the part ii let us take u JF. t Then there exists f SN 2 F, λ P such that u = fsdbs. Thus by Doob s maximal inequality we have v E u 2 E sup fsdbs 2 2E v [,t] fsdbs 2 = [,t] Ω f 2 ds dp

5 FUZZY STOCHASTIC INTEGRAL EQUATIONS 477 I Ω F 2 ds dp. Thus we have that J t F is a bounded subset of L2 Ω, A t, P; R d for, t I, < t. Under the assumptions, the set SN 2 F, λ P is also norm-closed. Moreover, from the above estimations it follows that the sets SN 2 F, λ P and Jt F are bounded in reflexive spaces L 2 N λ P and L2 Ω, A t, P; R d, respectively. Thus the set J tf is conditionally weakly compact and SN 2 F, λ P is weakly compact see [1]. Let {u n } JF t and let us suppose u n u in L 2 Ω, A t, P; R d. Then there exist a sequence {f n } SN 2 F, λ P such that u n = f nsdbs for n N. Hence, by Itô λ P. Thus isometry for Itô integral, we infer that {f n } is a Cauchy sequence in L 2 N it has to be convergent to some element f L 2 N λ P. Since the set S2 N F, λ P is closed, we conclude that u = fsdbs. This proves the closedness of the set F and consequently its weak compactness. J t Remark 2.3. If F : I Ω Kc brd is a nonanticipating and L 2 N λ P-integrably bounded set-valued stochastic process, then SN 2 F, λ P and Jt F are convex sets. Theorem 2.4. For each n N, let F n : I Ω K b R d be a nonanticipating setvalued stochastic processes such that F 1 is L 2 N λ P-integrably bounded and where F := F n n=1 F 1 F 2... F λ P-a.e., λ P-a.e. Then for every, t I, < t it holds JF t = JF t n. n=1 Proof. Firstly, let us note that by Theorem 3.3 in [21] a set-valued mapping F is nonanticipating and L 2 N λ P-integrably bounded. Since SN 2 F 1, λ P SN 2 F 2, λ P... SN 2 F, λ P, we infer that S 2 N F, λ P f n=1 n=1 S 2 N F n, λ P. Let us suppose that there exists S 2 N F n, λ P \ S 2 N F, λ P. Then for every n N it holds f F n λ P-a.e., and consequently f F λ P-a.e. Since f L 2 N λ P, it has to belong to the set SN 2 F, λ P too. This leads to the contradiction. Hence SN 2 F, λ P = SN 2 F n, λ P. Thus for every fixed, t I, < t we have n=1 { { n=1 } fsdbs : f SNF, 2 λ P = } fsdbs : f SN 2 F n, λ P,

6 478 M. T. MALINOWSKI AND M. MICHTA what completes the proof. Theorem 2.5. Let F 1, F 2 : I Ω K b R d be the nonanticipating and L 2 N λ P- integrably bounded set-valued stochastic processes. Then for every, t I, < t it holds H 2 L 2 J t F 1, J t F 2 [,t] Ω H 2 R d F 1, F 2 ds dp. Proof. Let us fix, t I such that < t. Then, for a fixed selection f 1 S 2 N F 1, λ P let us define a function ϕ: I Ω R d R + by ϕt, ω, x := f 1 t, ω x 2. Then ϕ,, x is N-measurable for every fixed x R d and the mapping ϕt, ω, is continuous for every fixed t, ω I Ω. Hence by Theorem 2.2 in [13] we obtain dist 2 L f 2 1 sdbs, JF t 2 = inf f 2 S 2 N F 2,λ P = inf = f 2 SN 2 F 2,λ P [,t] Ω [,t] Ω [,t] Ω f 1 sdbs f 2 sdbs ϕs, ω, f 2 s, ωds dp inf ϕs, ω, xds dp x F 2 s,ω H 2 R d F 1, F 2 ds dp. 2 L 2 Therefore we claim that sup dist 2 L 2 j1, JF t 2 j 1 JF t 1 In a similar way one can show that sup dist 2 j 2 J tf L 2 j2, JF t 1 2 [,t] Ω [,t] Ω H 2 R d F 1, F 2 ds dp. H 2 R d F 1, F 2 ds dp. Lemma 2.6. Let F : I Ω K b R d be a nonanticipating and L 2 N λ P-integrably bounded set-valued stochastic process. Then for every, a, t I such that a t it holds 2.1 J t F = J a F + J t af.

7 FUZZY STOCHASTIC INTEGRAL EQUATIONS 479 Proof. It is clear that J tf Ja F + Jt a F. We want to show that the opposite inclusion holds. Let us fix the random variables j 1, j 2 that belong to J a F and JaF, t respectively. Then there exist f 1, f 2 SN 2 F, λ P such that j 1 = a f 1sdBs, j 2 = f a 2sdBs. Let us take any f 3 SN 2 F, λ P. Since S2 N F, λ P is decomposable with respect to σ-algebra N, we infer that fs, ω = f 1 s, ω1 [,a] Ω s, ω + f 2 s, ω1 a,t] Ω s, ω + f 3 s, ω1 t,t] Ω s, ω belongs to SN 2 F, λ P, too. Observe that j 1 + j 2 = fsdbs. This completes the proof. Application of 2.1 together with Theorem 2.5 yields the following result. Theorem 2.7. Let F : I Ω K b R d be a nonanticipating and L 2 N λ P-integrably bounded set-valued stochastic process. Then the mapping [, T] t J t F Kb L 2 is continuous with respect to the metric H L 2. Now we consider the set-valued trajectory Aumann stochastic integral. Similarly as in the preceding considerations, let F : I Ω K b R d be a nonanticipating and L 2 N λ P-integrably bounded set-valued stochastic process. Then for, t I, < t we define set-valued trajectory Aumann stochastic integral L t F as a subset of L 2 Ω, A t, P; R d and described by L t F := { Then for every f SN 2 F, λ P one has E fsds 2 t fsds : f S 2 N F, λ P }. [,t] Ω f 2 ds dp. Proposition 2.8. Let F : I Ω K b R d be a nonanticipating and L 2 N λ P- integrably bounded set-valued stochastic process. Then set-valued trajectory stochastic integral L t F is nonempty, bounded, closed and weakly compact subset of L2 Ω, A t, P; R d. Proof. The boundedness property follows by the inequality above. The conditional weak compactness follows as in the proof of Proposition 2.2. Let {u n } L t F and let u n u in L 2 Ω, A t, P; R d. Then there exists a sequence {f n } SN 2 F, λ dp such that u n = f nsds. Since the set SN 2 F, λ P is weakly compact, there exists a subsequence {f nk } of {f n } such that f nk f in L 2 N λ P, where denotes weak convergence. But SN 2 F, λ P is weakly closed, thus f S2 N F, λ P. For a fixed

8 48 M. T. MALINOWSKI AND M. MICHTA, t I, < t, let us consider a linear operator T t : L2 N λ P L2 Ω, A t, P; R d defined by T t g := gsds. By the inequality above, it follows that T t is normto-norm continuous. Hence by Dunford Schwartz Theorem see [1] it is equivalent to the fact that T t is continuous with respect to weak topologies. Therefore u n k = T t f nk T t f in L 2 Ω, A t, P; R d, and hence u = T t f. This proves the closedness of the set L t F in the norm topology of the space L2 Ω, A t, P; R d, and consequently also its weak compactness. Remark 2.9. If F : I Ω Kc brd is a nonanticipating and L 2 N λ P-integrably bounded set-valued stochastic process, then L t F is a convex set. Clearly we have also that for every, a, t I, a t it holds L t F = L a F + L t af. Moreover, if F 1, F 2 : I Ω K b R d are the nonanticipating and L 2 N λ P- integrably bounded set-valued stochastic processes, then for every, t I, < t, similarly as in the proof of Theorem 2.5, one can derive the following: for every f 1 SN 2 F 1, λ P we have dist 2 L f 2 1 sds, L t F 2 = inf E f 2 SN 2 F 2,λ P t = t t inf f 2 SN 2 F 2,λ P [,t] Ω [,t] Ω [,t] Ω f 1 s f 2 sds 2 f 1 s f 2 s 2 ds dp inf ϕs, ω, xds dp x F 2 s,ω H 2 R d F 1, F 2 ds dp, where ϕ: I Ω R d R + is defined as ϕs, ω, x := f 1 s, ω x 2. Hence we have the following result. Theorem 2.1. Let F 1, F 2 : I Ω K b R d be the nonanticipating and L 2 N λ P- integrably bounded set-valued stochastic processes. Then for every, t I, < t it holds H 2 L 2 L t F 1, L t F 2 t [,t] Ω H 2 R d F 1, F 2 ds dp. Remark Similarly as in Theorem 2.7 one obtains that the mapping is continuous with respect to H L 2. [, T] t L t F Kb L 2

9 FUZZY STOCHASTIC INTEGRAL EQUATIONS 481 Finally, using the same arguments as in the proof of Theorem 2.4 one can show the following result. Theorem For each n N, let F n : I Ω K b R d be a nonanticipating set-valued stochastic processes such that F 1 is L 2 N λ P-integrably bounded and where F := F n n=1 F 1 F 2 F λ P-a.e., λ P-a.e. Then for every, t I, < t it holds L t F = L t F n. n=1 3. FUZZY TRAJECTORY STOCHASTIC INTEGRALS AND FUZZY STOCHASTIC INTEGRAL EQUATION As it was announced in the Introduction, in this section we recall some fundamental facts concerning fuzzy sets and fuzzy stochastic processes. Then we formulate the results of existence of fuzzy trajectory stochastic integrals of Itô and Aumann type. Finally we consider the fuzzy stochastic integral equations with the fuzzy trajectory stochastic integrals. Let X be a given separable, reflexive Banach space. By a fuzzy set u of the space X we mean a function u: X [, 1]. We denote this fact as u FX. For α, 1] denote [u] α := {x X : ux α} and let [u] := cl X {x X : ux > }, where cl X denotes the closure in X, X. The sets [u] α are called the α-level sets of fuzzy set u, and -level set is called the support of u. We will use the following Representation Theorem of Negoita Ralescu [36]. Theorem 3.1. Let M X be a set and let {C α : α [, 1]} be a family of subsets of M such that i C = M, ii C C α C β for α β, iii if α n ր α then C α = C αn. n=1 Then there exists u FX such that [u] α = C α for every α [, 1]. Moreover sup{α : x C α }, if x M, ux =, if x M. In the sequel we will deal with fuzzy sets which have some additional properties. Therefore we introduce the notation F b X = {u FX : [u] α K b X for every α [, 1]},

10 482 M. T. MALINOWSKI AND M. MICHTA F b cx = {u F b X : [u] α K b cx for every α [, 1]}. The following metric D X in F b X is often used: D X u, v := sup H X [u] α, [v] α for u, v F b X. α [,1] This metric generalizes the Hausdorff metric H X, and has the property D X u 1 + u 2, v 1 + v 2 D X u 1, v 1 + D X u 2, v 2 for u 1, u 2, v 1, v 2 F b c X, where the addition of fuzzy sets is defined levelwise, i.e. [u 1 + u 2 ] α = [u 1 ] α + [u 2 ] α for α [, 1]. It is known see [43] that F b X, D X is a complete metric space. It is also easy to see that F b c X is a closed subset of Fb X. For our aims we will consider two cases of X. Namely we will take X = R d or X = L 2, where we assume from now on that σ-algebra A is separable with respect to probability measure P. By a fuzzy random variable we mean a function u: Ω F b X such that [u ] α : Ω K b X is an A-measurable set-valued mapping for every α, 1]. A fuzzy set-valued mapping f : I Ω F b X is called a fuzzy stochastic process if ft, : Ω F b X is a fuzzy random variable for every t I. The fuzzy stochastic process f : I Ω F b X is said to be nonanticipating if the set-valued mapping [f] α : I Ω K b X is N-measurable for every α, 1]. A fuzzy stochastic process f is called {A t }-adapted, if ft, is an A t -measurable fuzzy random variable for every t I. Let f : I Ω F b X be a nonanticipating fuzzy stochastic process. The process f is said to be L 2 N λ P-integrably bounded, if [f] L 2 N λ P. Consider now f : I Ω F b R d and assume that it is nonanticipating and L 2 N λ P-integrably bounded fuzzy stochastic process. For α-levels α [, 1] of such a fuzzy stochastic process f one can consider set-valued trajectory stochastic integrals J t [f] α and L t [f] α for every, t I, < t. Then by Theorem 2.4, Theorem 2.12 and Theorem 3.1 for every, t I, < t there exist fuzzy sets in F b L 2 denoted by F fsdbs and F such that for every α [, 1] [ ] α F fsdbs = J[f] t α, [ F fsds fsds ] α = L t [f] α.

11 Definition 3.2. The fuzzy sets F FUZZY STOCHASTIC INTEGRAL EQUATIONS 483 fsdbs F b L 2, F fsds F b L 2 described above are called the fuzzy trajectory stochastic Itô integral and fuzzy trajectory stochastic Aumann integral respectively of nonanticipating and L 2 N λ P- integrably bounded fuzzy stochastic process f : I Ω F b R d. Note that the sum of these integrals not necessarily is the fuzzy set from F b L 2. Remark 3.3. If we assume additionally that f has convex α-level sets, i.e. f : I Ω F b c Rd then F and as a consequence F fsdbs F b c L2, F fsds + F This fact we will use later in 3.3. fsds F b c L2, fsdbs F b cl 2. One can show that for every, a, t I, a t it holds 3.1 F 3.2 F fsdbs = F fsds = F a a fsdbs + F fsds + F a a fsdbs, fsds. Directly by Theorem 2.5 and Theorem 2.1 we have the following result. Corollary 3.4. Let f, g: I Ω F b R d be a nonanticipating and L 2 N λ P- integrably bounded fuzzy stochastic processes. Then for every, t I, < t D 2 L F fsdbs, F gsdbs D 2 2 R f, gds dp, d and D 2 L 2 F fsds, F Applying 3.1 and 3.2 we obtain: gsds t [,t] Ω [,t] Ω D 2 R d f, gds dp. Corollary 3.5. Let f : I Ω F b R d be a nonanticipating and L 2 N λ P-integrably bounded fuzzy stochastic process. Then the mappings [, T] t F fsdbs F b L 2,

12 484 M. T. MALINOWSKI AND M. MICHTA [, T] t F fsds F b L 2, are continuous with respect to the metric D L 2. In the sequel we want to consider some fuzzy equation in which the fuzzy trajectory stochastic integrals appear. Denote L 2 = L2 Ω, A, P; R d. Let f, g: I Ω Fc bl2 Fc brd and let X FcL b 2. By a fuzzy stochastic integral equation we mean the following relation in the space Fc bl2 : 3.3 Xt = X + F fs, Xsds + F gs, XsdBs for t I. Definition 3.6. By a solution to 3.3 we mean a continuous mapping X : I Fc bl2 that satisfies 3.3. A solution X : I Fc bl2 to 3.3 is unique if Xt = Y t for every t I, where Y : I Fc bl2 is any solution of 3.3. Below we write down the detailed conditions which will be imposed on the coefficients of the equation. Assume that f, g: I Ω FcL b 2 FcR b d satisfy: f1 for every u Fc bl2 the mappings f,, u, g,, u: I Ω Fc b Rd are the nonanticipating fuzzy stochastic processes, f2 there exists a constant K > such that D R d ft, u, ft, v + DR d gt, u, gt, v KDL 2u, v for every t, ω I Ω, and every u, v Fc bl2, f3 there exists a constant C > such that D R d ft, u, ˆθ + DR d gt, u, ˆθ C 1 + DL 2u, ˆΘ, for every t, ω I Ω, and every u FcL b 2. To describe the symbols ˆθ, ˆΘ let θ, Θ denote the zero elements in R d and L 2, respectively. Then the symbols ˆθ, ˆΘ are their fuzzy counterparts, i.e. [ˆθ] α = {θ} and [ˆΘ] α = {Θ} for every α [, 1]. Theorem 3.7. Let X Fc bl2, and f, g: I Ω Fb c L2 Fc brd satisfy the conditions f1-f3. Then the equation 3.3 has a unique solution.

13 FUZZY STOCHASTIC INTEGRAL EQUATIONS 485 Proof. Let us define a sequence X n : I F b c L2, n =, 1,... of successive approximations as follows: and for n = 1, 2,... for every t I. X n t = X + F X t = X, for every t I, fs, X n 1 sds + F Due to Corollary 3.5, the mappings X n are continuous. gs, X n 1 sdbs By Corollary 3.4 and assumptions f1, f3 one gets DL F 2 fs, X 2 ds, ˆΘ = D 2L F fs, X 2 ds, F ˆθds T D 2 R fs, d X, ˆθ ds dp ηt 2, I Ω where η = 2C D 2 L 2 X, ˆΘ. Similarly we have D 2 L 2 F Therefore for every t I gs, X dbs, ˆΘ = D 2L F 2 I Ω D 2 L 2 X1 t, X t 2D 2 L 2 F gs, X dbs, F ˆθds D 2 R d gs, X, ˆθ ds dp ηt. + 2D 2 L 2 F 2ηTT + 1. fs, X ds, ˆΘ gs, X dbs, ˆΘ Observe further that for n = 2, 3,..., using Corollary 3.4 and assumptions f1-f2, one has D 2 L 2 Xn t, X n 1 t 2D 2 L 2 F fs, X n 1 sds, F fs, X n 2 sds + 2D 2 L F gs, X 2 n 1 sdbs, F 2T D 2 R fs, d Xn 1 s, fs, X n 2 s ds dp gs, X n 2 sdbs I Ω + 2 D 2 R d gs, Xn 1 s, gs, X n 2 s ds dp I Ω

14 486 M. T. MALINOWSKI AND M. MICHTA T Thus we obtain 2K 2 T + 1 sup D 2 L 2 Xn 1 s, X n 2 s ds. D 2 L 2 Xn t, X n 1 t ηk 2[2K2 TT + 1] n. n! Let us consider the space CI, F b c L2 of continuous mappings X : I F b c L2 with a distance ρ defined by ρx, Y = sup D L 2Xt, Y t for X, Y CI, FcL b 2. It is clear that CI, F b c L2, ρ is a complete metric space. Since {X n } CI, F b c L2 and for m < n ρx n, X m = sup D L 2X n t, X m t ηk 1 n k=m+1 2K2 TT + 1 k, k! we infer that {X n } n= is a Cauchy sequence in CI, F b c L2, ρ. Thus there is X CI, F b cl 2 such that ρx n, X, as n. We shall show that X is a solution to 3.3. Let t I be fixed. We have D 2 L Xt, X 2 + F fs, Xsds + F gs, XsdBs 3D 2 L 2 Xn t, Xt + 3D 2 L 2 X n t, X + F + F + 6D 2 L 2 F + 6D 2 L 2 F gs, X n 1 sdbs fs, X n 1 sds fs, X n 1 sds, F gs, X n 1 sdbs, F fs, Xsds gs, XsdBs, The first term on the right-hand side of the inequality converges to zero, whereas the second is equal to zero. Since and D 2 L 2 F D 2 L 2 F fs, X n 1 sds, F T TK 2 DL 2 2 Xn 1 s, Xs ds, fs, Xsds T 2 K 2 sup D 2 L 2 Xn 1 t, Xt, as n, gs, X n 1 sdbs, F gs, XsdBs

15 we infer that Xt, X + F D L 2 for every t I. FUZZY STOCHASTIC INTEGRAL EQUATIONS 487 T K 2 D 2 L 2 Xn 1 s, Xs ds, TK 2 sup D 2 L 2 Xn 1 t, Xt, as n, fs, Xsds + F gs, XsdBs = For the uniqueness assume that X : I F b cl 2 and Y : I F b cl 2 are two solutions to 3.3. Then let us notice that D 2 L 2 Xt, Y t 2K 2 T + 1 Thus, by Gronwall s lemma, we obtain D 2 L 2 Xs, Y s ds. D 2 L 2 Xt, Y t for every t I. Therefore the uniqueness of the solution follows. The next result presents some estimation for the solution to 3.3. Theorem 3.8. Let X F b c L2 and f, g: I Ω Fb c L2 F b c Rd satisfy the assumptions of Theorem 3.7. Then the solution X of 3.3 satisfies sup D 2 L Xt, ˆΘ 3D 2 2 L 2 X, ˆΘ + 6C 2 TT + 1 e 6C2TT+1. Proof. Let us fix t I. Using Corollary 3.4 and assumption f3 we obtain D 2 L Xt, ˆΘ 3D 2 2 L 2X, ˆΘ t + 3D 2 L F fs, Xsds, ˆΘ 2 t + 3D 2 L F gs, Xsds, ˆΘ 2 3D 2 L 2X, ˆΘ + 3t D 2 R d fs, Xs, ˆθ ds dp + 3 [,t] Ω D 2 R d gs, Xs, ˆθ ds dp [,t] Ω 3D 2 L 2X, ˆΘ + 6C 2 tt C 2 t + 1 Now it is enough to apply the Gronwall lemma to get the result. D 2 L 2 Xs, ˆΘ ds Next we will show a continuous dependence on initial conditions of the solution to 3.3. Let X, Y denote the solutions of the equations 3.4 Xt = X + F fs, Xsds + F gs, XsdBs for t I,

16 488 M. T. MALINOWSKI AND M. MICHTA 3.5 Y t = Y + F fs, Y sds + F gs, Y sdbs for t I, respectively. Assume that X, Y and f, g satisfy the conditions as in Theorem 3.7. Theorem 3.9. For the solutions X, Y of the equations 3.4, 3.5 it holds sup D 2 L 2Xt, Y t 3D2 L 2X, Y e 3K2TT+1. Proof. It is enough to notice that for every t I one has D 2 L 2 Xt, Y t 3D 2 L 2X, Y + 3t D 2 R d fs, Xs, fs, Y s ds dp + 3 [,t] Ω D 2 R d gs, Xs, gs, Y s ds dp [,t] Ω 3D 2 L 2X, Y + 3K 2 t + 1 D 2 L 2 Xs, Y s ds. Now application of the Gronwall lemma gives the result. Finally we present the stability property of solutions to the system of fuzzy stochastic integral equations. Let us consider the following equations: Xt = X + F and for n = 1, 2,... for t I. X n t = X,n + F fs, Xsds + F gs, XsdBs for t I, f n s, X n sds + F g n s, X n sdbs Theorem 3.1. Let f, g, f n, g n : I Ω F b cl 2 F b cr d satisfy the conditions f1-f3 with the same constants K, C. Let also X, X,n F b cl 2 for every n N. Assume that i D L 2 X,n, X, ii D R d fn t, u, ft, u, for every t, ω, u I Ω F b c L2, iii D R d gn t, u, gt, u, for every t, ω, u I Ω F b c L2. Then sup D L 2 Xn t, Xt, as n.

17 FUZZY STOCHASTIC INTEGRAL EQUATIONS 489 Proof. By virtue of Corollary 3.4, assumptions f1-f3, let us note that for every t I one can show the following estimations: D 2 L 2 Xn t, Xt 3D 2 L 2 X,n, X + 3t D 2 R d fn s, X n s, fs, Xs ds dp + 3 [,t] Ω D 2 R d gn s, X n s, gs, Xs ds dp [,t] Ω 3D 2 L 2 X,n, X + 6t D 2 R d fn s, Xs, fs, Xs ds dp + 6 [,t] Ω D 2 R d gn s, Xs, gs, Xs ds dp [,t] Ω + 6K 2 t + 1 D 2 L 2 Xn s, Xs ds. Application of Gronwall s lemma yields D 2 L 2 Xn t, Xt 3D 2L 2 X,n, X + 6t D 2 R d fn s, Xs, fs, Xs ds dp + 6 [,t] Ω D 2 R d gn s, Xs, gs, Xs ds dp e 6K2 tt+1. [,t] Ω Using the assumptions and the Lebesgue Dominated Convergence Theorem we end the proof. 4. SET-VALUED STOCHASTIC INTEGRAL EQUATIONS In this section we consider set-valued stochastic integral equations. The motivation of the study of such equations comes from the deterministic case, where the study of them has been used as an alternative approach to the issue of fuzzy differential equations see e.g. [26, 28]. The idea used there was to repleace an original fuzzy differential equation by the system of set-valued differential equations generated from the original fuzzy setup. Such procedure avoided the main disadvantage of studying directly the fuzzy differential equation in the original formulation, i.e, the lack of a reflection of the rich behaviour of corresponding differential equations without fuzziness. As it was shown, it had been caused by the fact that the diameter of any solution of a fuzzy differential equation increases as time increases because of the necessity of the fuzzification of the derivative involved. Taking these remarks into consideration, let F, G: I Ω Kc bl2 Kc brd and let X Kc bl2. By a

18 49 M. T. MALINOWSKI AND M. MICHTA set-valued stochastic integral equation we mean the following relation in the space K b cl 2 : 4.1 Xt = X + L t FX + Jt GX for t I, where FX, GX: I Ω K b c Rd are defined by FXt, ω = Ft, ω, Xt, GXt, ω = Gt, ω, Xt for t, ω I Ω. Definition 4.1. By a solution to 4.1 we mean a continuous mapping X : I K b c L2 that satisfies 4.1. A solution X : I K b c L2 to 4.1 is unique if Xt = Y t for every t I, where Y : I K b c L2 is any solution of 4.1. Now we formulate the conditions required from the equation coefficients. Assume that F, G: I Ω K b cl 2 K b cr d satisfy: s1 for every u K b cl 2 the mappings F,, u, G,, u: I Ω K b cr d are the nonanticipating set-valued stochastic processes, s2 there exists a constant K > such that H R d Ft, u, Ft, v + HR d Gt, u, Gt, v KHL 2u, v for every t, ω I, and every u, v K b c L2, s3 there exists a constant C > such that H R d Ft, u, θ + HR d Gt, u, θ C 1 + HL 2u, Θ, for every t, ω I Ω, and every u K b c L2. Using arguments which are similar to those from Section 3, we obtain the results for set-valued stochastic integral equations. Theorem 4.2. X K b cl 2, and F, G: I Ω K b cl 2 K b cr d satisfy the conditions s1-s3. Then equation 4.1 has a unique solution. Theorem 4.3. Let X K b cl 2 and F, G: I Ω K b cl 2 K b cr d satisfy the assumptions of Theorem 4.2. Then the solution X of 4.1 satisfies sup H 2 L Xt, Θ 3H 2 2 L 2 X, Θ + 6C 2 TT + 1 e 6C2TT+1. Also a continuous dependence on initial conditions of solution to 4.1 holds: let X, Y denote the solutions of the equations 4.2 Xt = X + L t FX + Jt GX for t I, 4.3 Y t = Y + L t FY + Jt GY for t I,

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