Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

Abstract We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.


Introduction
Mathematical economics and optimal control are typical fields widely using the methods of the set-valued analysis (see e.g. [10,27]). The set-valued mappings are fundamental for differential inclusions (cf. [7,19,29,52]) and for certain research areas in physics, including theory of defects in crystal, magnetic monopoles, vortices in superfluids and superconductors (see e.g. [22]). Differential inclusions and fuzzy differential inclusions [6], a generalization of differential equations, find their application in dynamical systems with incomplete information and velocities that are not uniquely determined by the state of the considered system. These can also be used for modeling systems with uncertainty resulting from vagueness of human knowledge (for example, it very often happens that the perfect value of the initial condition is not known with only the set of initial values being determined). However, the real-world phenomena characterized by uncertainty are often subjected to random forces. Hence, the deterministic differential inclusions or stochastic differential equations cannot always serve as tools adequate enough to model such phenomena. In such situations stochastic differential inclusions can be helpful. Stochastic differential inclusions combine two ways of representing uncertainty: a stochastic uncertainty generated by random noise and a contingent uncertainty driven by set-valued mappings (see e.g. [5, 8, 9, 11, 12, 14, 17, 20, 28, 30, 45, 49-51, 53, 56-58]).
In this paper we focus on a slightly different new approach that can be useful in dealing with phenomena subjected to stochastic and contingent uncertainties. It involves the so-called set-valued stochastic integral equations (SSIEs, for short) [38][39][40][41][42]. Such equations have solutions that are set-valued mappings taking on values in the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L 2 (consisting of square integrable random vectors). In contrast, the stochastic differential inclusions have solutions that are single-valued stochastic processes. However, SSIEs and stochastic differential inclusions both generalize ordinary stochastic differential equations. SSIEs extend the notion of the deterministic set-valued differential equations [1-4, 15, 18, 21, 24, 25, 33, 35-37, 55] that have coalesced into an independent research discipline [31]. SSIEs are also a basis for studies of the fuzzy stochastic integral equations [38][39][40][41][42].
In [38][39][40][41][42] both the existence and uniqueness of solutions to SSIEs are proved under the Lipschitz-type conditions. In this paper, given the Osgood type condition (for which the Lipschitz condition as stated in [38][39][40][41][42] is a special case), we perform an analysis of the problem of existence and uniqueness of solutions to SSIEs driven by semimartingales. In order to obtain the continuity of solutions we assume that the integrators have continuous sample paths. We show that solutions to SSIEs depend continuously on the data of the equation. To avoid future repetitions, we also investigate the problem of possible generalizations of SSIEs. Thus we arrive at the fuzzy stochastic integral equations under the Osgood type condition.
The paper is organized as follows. In Section 2 we recall the definitions and properties of the set-valued stochastic trajectory integrals with respect to the processes of finite variation and martingales. Section 3 is devoted to the analysis of SSIEs with two stochastic trajectory integrals with respect to processes A; M (respectively) from decomposition Z D A C M of semimartingale Z. In Section 4 we consider SSIEs with only one integral driven by semimartingale Z. The integrands F; G in Section 3 come from wider classes than the integrand F in Section 4, but integrator Z in Section 4 is more general than integrators A; M in Section 3. The equations considered in Sections 3 and 4 are not equivalent. Although the results in Sections 3 and 4 are similar we decided to present them separately, since these can be viewed as independent research areas. In Section 5 and 6 we describe the results concerning fuzzy stochastic integral equations with coefficients satisfying the Osgood type condition.

Preliminaries
Let .X ; k k X / be a separable Banach space, K b c .X / the family of all nonempty, closed, bounded and convex subsets of X . The Hausdorff metric H X in K b c .X / is defined by where dist X .a; B/ WD inf b2B ka bk X . It is known (cf. [26]) that .K b c .X /; H X / is a complete and separable metric space. Also, the set K b c .X / has a semilinear structure under addition and scalar multiplication defined as: For nonempty subsets A 1 ; A 2 ; B 1 ; B 2 of X it holds Let .U; U; / be a measure space. A set-valued mapping (multifunction) F W U ! K b c .X / is said to be measurable if it satisfies: c .X / is said to be L p U . /-integrally bounded (p > 1), if there exists h 2 L p .U; U; I R/ such that the inequality H X .F; fÂ X g/ 6 h holds -a.e., where Â X denotes the zero element in X . It is known (see [23]) that F is L p U . /-integrally bounded if, and only if, u 7 ! H X .F .u/; fÂ X g/ belongs to L p .U; U; I R/.
Denote I D OE0; T , where T < 1. Let .˝; A; fA t g t 2I ; P / be a complete filtered probability space satisfying the usual hypotheses, i.e. fA t g t 2I is an increasing and right-continuous family of sub--algebras of A and A 0 contains all P -null sets. We will also assume that -algebra A is separable with respect to probability measure P . Let P denotes the -algebra of predictable elements in I ˝, i.e. the smallest -algebra with respect to which every fA t g-adapted stochastic process with left-continuous sample paths is measurable. A stochastic process Let ZW I ˝! R be a semimartingale with the canonical representation where AW I ˝! R is an fA t g-adapted cádlág stochastic process of finite variation, M W I ˝! R is a local fA t gmartingale. It is known that if Z has continuous sample paths then A; M have continuous sample paths, too. Also A is predictable and representation (1) is unique. Obviously, each process A and M can be treated as a semimartingale. Since A is of finite variation, almost each (w.r.t. to P ) sample path A. ; !/W I ! R generates a measure A. ;!/ with the total variation on the interval OE0; t given by jA.!/j t D R t 0 A. ;!/ .ds/. For a local martingale M one can define the quadratic variation process OEM W I ˝! R (cf. [16]). Now we denote by H 2 the set of all semimartingales where L 2 WD L 2 .˝; A; P I R d /. For further aims we denote also L 2 t WD L 2 .˝; A t ; P I R d /, t 2 I . It is known (cf. [48] chap. II, sec. 6, Corollary 4) that for a continuous semimartingale Z 2 H 2 the process M in (1) is a continuous square integrable martingale and EjAj 2 T < 1. The processes A; M from the representation (1) of the semimartingale Z induce two measures A ; M defined on .I ˝; P/. The measure A is defined similarly as in [14] A .C / WD For f 2 L 2 P . A /, where L 2 P . A / WD L 2 .I ˝; P; A I R d / one can define the single-valued stochastic Lebesgue-Stjeltjes integral R t 0 f .s/dA.s/ sample path by sample path (cf. [48]). Note that The second measure M is the well-known Doléan-Dade measure (cf. [16]), i.e. a measure such that and t 2 I one can define the single-valued stochastic integral R t 0 f .s/dM s and we have (cf. [16]) Let F; GW I ˝! K b c .R d / be some predictable set-valued stochastic processes. We will assume throughout the paper that F is L 2 P . A /-integrally bounded and G is L 2 P . M /-integrally bounded. For such processes let us define the sets Due to the Kuratowski and Ryll-Nardzewski Selection Theorem we have S 2 P .F; A / 6 D ;, S 2 P .G; M / 6 D ;, and we can define the set-valued stochastic trajectory integrals (see [41]).

Definition 2.2.
(a) For a predictable and L 2 P . A /-integrally bounded set-valued stochastic process F W I ˝! K b c .R d / and for ; t 2 R C , < t the set-valued stochastic trajectory integral (over interval OE ; t ) of F with respect to the bounded variation process A is the following subset of For a predictable and L 2 P . M /-integrally bounded set-valued stochastic process GW I ˝! K b c .R d / and for ; t 2 R C , < t the set-valued stochastic trajectory integral (over interval OE ; t ) of G with respect to the martingale M is the following subset of L 2 It is known (see [41]) that R t F .s/dA.s/ and R t G.s/dM.s/ are nonempty, bounded, convex, closed and weakly compact subsets of L 2 t . We will exploite the following properties. (iii) for every 2 I the mappings are H L 2 -continuous.

Set-valued equations driven by processes A, M from decomposition of semimartingale Z
We begin this section with a discussion on potential utility of the theory of set-valued stochastic integral equations.
We indicate that such equations can be used in modeling dynamics of real-world phenomena subjected to uncertainty and also in modeling optimality problems. Consider a situation when a pharmacist grows a population of a bacteria species that lives on a given bounded territory. Assume that the number of individuals is affected by random factors and the pharmacist can control the growth of the population by steerage of feeding. Then the number of individuals at the instant t 2 I , denoted by n.t/, is random and could be modelled by the controlled stochastic integral equation where n 0 W˝! R denotes initial number of individuals, f W R 2 ! R symbolizes drift coefficient, gW R 2 ! R is a diffusion coefficient, u denotes a strategy of feeding, u 2 U , U is a set of feeding actions -controls, W denotes the Wiener process. Assuming that n.t / 2 L 2 for t 2 I we can transform (4) to an equation in the space L 2 , i.e. to the equation where the mappings N f ; N gW I ˝ L 2 U ! R are defined as follows N f .s; !; n; u/ WD f .n.!/; u.s; !// and N g.s; !; n; u/ WD g.n.!/; u.s; !//: In most situations the pharmacist is not able to determine n 0 precisely. Suppose that it is only known that n 0 is an A 0 -measurable random variable with values bounded by a constant > 0. In this way, under presence of uncertainty, the initial number of individuals can be viewed as the following set The uncertain number of individuals is governed by dynamics of set-valued solution to the set-valued stochastic integral equation The symbol co.B/ denotes the closed and convex hull of the set B.
The equation (6) is a set-valued stochastic integral equation driven by processes from decomposition of semimartingle Z D A C M with A.s/ D s and M.s/ D W .s/. A problem of finding its solution is a natural question. The set-valued solution can give informations on an approximate dynamics of population growth.
Having equations (5) and (6) one can treat the continuous solution nW I ! L 2 as a mapping which satisfy n.t/ 2 X.t/ for t 2 I , where X denotes set-valued solution to (6). This means that n is a continuous selection of X . The pharmacist can associate a cost˚.n/ to the achieved solution n. Now, it arises a question of lowest value of the cost. If such value exists one can ask about existence of a solution which realizes the lowest value of the cost. This leads us to an optimality problem. More precisely, we obtain a problem of existence of a continuous selection N n of set-valued solution X to (6) with the property˚. where the infimum ranges over all continuous selections of the solution to (6). Hence the problem of existence and uniqueness of a solution to (6) is fundamental.
In this section we consider a notion of an SSIE driven by processes A; M from decomposition (1) of semimartingale Z. Such equation is understood as the following relation considered in the metric space .K b c .L 2 /; H L 2 /: where Definition 3.1. An H L 2 -continuous mapping X W I ! K b c .L 2 / is called the solution to (7) if X satisfies (7) and Throughout the paper we will assume that A, M are such that A , M are absolutely continuous with respect to the Lebesgue measure , so that the Radon-Nikodem derivatives exist. Also we will assume that Hence in this case the assumptions written above are fulfiled.
For our further aims we denote possesses these properties as well.
In this part of the paper we shall show the existence and uniqueness theorem for solutions to (7) in the case when F and G satisfy the Osgood type condition. Therefore we formulate the following assumptions: there exists a continuous, nondecreasing, concave function ÄW OE0; 1/ ! OE0; 1/ satisfying: The assumption (A3) is called the Osgood type condition and it is weaker than the Lipschitz condition imposed in [38][39][40][41][42].
where M is a positive constant, then condition (A3) reduces to the Lipschitz condition.
In the derivation of existence of solution to (7) we will use the sequence fX n g 1 nD1 of approximate solutions X n W I !
for t 2 k 1 n ; k n i \ I , k D 1; 2; : : : Note that each approximant X n appearing above is H L 2 -continuous and X n .
then we can write Before we formulate the main result concerning equation (7), we list some useful lemmata. We begin with Bihari's inequality which will play a key role in derivations of our results.
Lemma 3.5 ( [13]). Let f W I ! OE0; 1/ be continuous and wW I ! OE0; 1/ be continuous and nondecreasing, and let gW OE0; 1/ ! OE0; 1/ be continuous nondecreasing function such that g.s/ > 0 for s > 0. If f satisfies the following integral inequality where c is a positive constant, then and J 1 is the inverse function of J .
In our setting the role of function g, which appears in Lemma 3.5, will be played by the function Ä from assumption (A3). Then it is easy to observe that in our setting we have J.r/ Proof. For n 2 N and t 2 I we have, due to Proposition 2.3 (ii), Applying Remark 3.4 we get Hence for every n 2 N and every t 2 I Further we can infer that Using Lemma 3.5 we obtain . Then for every n 2 N and every r; t 2 I , r 6 t Proof. Due to Proposition 2.3 (i) and (ii), Remark 3.4, Lemma 3.6 we have: for n 2 N and r 6 t . Then for n 2 N and t 2 I where M 2 is as in Lemma 3.7.
Proof. Observe that for n; m 2 N and for t 2 I we have Since Ä is nondecreasing, using Corollary 3.8 we get Finally, by Bihari's inequality we infer that for every t 2 I sup w2OE0;t H 2 L 2 X n .w/; X m .w/ 6 J 1 J.A n;m / C 6 .t / 6 J 1 J.A n;m / C 6. A C M / : Since J 1 J.A n;m / C 6. A C M / n;m!1 ! 0, we get the assertion.
Now we are in a position to prove a main result of this section.
where M 1 is as in Lemma 3.6. Proof.
It is clear that C endowed with the supremum metric becomes a complete metric space.
Due to Lemma 3.9 the Maruyama sequence fX n g is a Cauchy sequence in C. Therefore there exists X ? 2 C such that sup t 2I H L 2 X n .t /; X ? .t / ! 0; as n ! 1: We shall show that X ? is the desired solution to (7). To this end let us note that for t 2 I we have Using Corollary 3.8 we obtain Now it is easy to see that the last expression converges to zero. Hence we obtain What is left is to prove that solution X ? is unique. Let us assume that X ? , Y ? are two solutions to (7). Then for every t 2 I it holds where " > 0. By Bihari's inequality for every t 2 I . Passing to the limit as " goes to zero, we obtain H 2 L 2 .X ? .t /; Y ? .t // D 0 for t 2 I . Now let us consider equation (7) . Then for the solutions X; Y of (7) and (8), respectively, the following estimation is true By Bihari's inequality, for every t 2 I we get This ends the proof.
Due to the above estimation we easily infer on continuous dependence of solution to (7) with respect to initial value.
Corollary 3.12. Let X 0 2 K b c .L 2 0 / and F; G satisfy (A1)-(A3). Let X denote the solution of (7) and for n 2 N let X n denote the solution of equation In the sequel, let us consider equation (7) and the equation where nonlinearities V and Q are different from F; G. The next result presents a bound for the distance of X and Y which are solutions to (7) and (9), respectively.
Theorem 3.13 allows for easy deduction on continuous dependence of solution to (7) with respect to the nonlinearities F and G.
Corollary 3.14. Let X 0 2 K b c .L 2 0 / and F; F n ; G; G n W I ˝ K b c .L 2 / ! K b c .R d / satisfy (A1)-(A3). Let X denote solution to (7) and X n denote solution to the equation H L 2 X n .t /; X.t / ! 0; as n ! 1: Let us notice that the results established for SSIEs have immediate consequences for single-valued stochastic integral equations driven by semimartingales [48]. Namely, consider f; gW I ˝ L 2 ! R d and x 0 2 L 2 0 . Then the following single-valued stochastic integral equation can be viewed as SSIE (7) with F D ff g, G D fgg and X 0 D fx 0 g. Therefore rewritting (A1)-(A3) in terms of f; g we obtain the following conditions

Set-valued equations driven by semimartingale Z
In this section we will need a notion of a set-valued stochastic integral with respect to semimartingales. As before, let ZW I ˝! R be a continuous semimartingale with the canonical representation (1). If additionally Z 2 H 2 , then one can define a finite measure Z on .I ˝; P/ as where the measures A ; M are as in Section 3. Let us denote Z WD Z .I ˝/; L 2 P . Z / WD L 2 .I ˝; P; Z I R d /: For f 2 L 2 P . Z / one can define the single-valued stochastic integral R t 0 f .s/dZ.s/ with respect to semimartingale Z as follows Due to (2) and (3) we claim that: Let F W I ˝! K b c .R d / be a predictable set-valued stochastic process which is L 2 P . Z /-integrally bounded. For such a process let us define the set S 2 P .F; Z / WD ff 2 L 2 P . Z / W f 2 F; Z -a.e.g. This set is nonempty, and similarly as e.g. in [38,42] we can define the set-valued stochastic trajectory integral with respect to semimartingales. c .R d / and for ; t 2 R C , < t the set-valued stochastic trajectory integral (over interval OE ; t ) of F with respect to the semimartingale Z is the following subset of L 2 It is known (cf. [38,42]) that R t F .s/dZ.s/ is a nonempty, bounded, convex, closed and weakly compact subset of Proceeding similarly as in preceding section we can obtain (without any difficulties) assertions which are parallels of those presented in Section 3. Now, in the derivations, we use the following properties.  (iii) for every 2 I the mapping OE ; An SSIE driven by semimartingale Z is the following relation in the metric space .K b c .L 2 /; H L 2 /: where X 0 2 K.
This equation is slightly different than (7). Here, the integrand Z is more general, but F comes from a narrower class than F in (7). Indeed, here F needs to be L 2 P . A /-integrally bounded and L 2 P . M /-integrally bounded, simultaneously, i.e. F needs to be L 2 P . Z /-integrally bounded.
An H L 2 -continuous mapping X W I ! K b c .L 2 / is called the solution to (11) if X satisfies (11) and X.t/ 2 K b c .L 2 t / for every t 2 I . A solution X is said to be unique if X.t / D Y .t / for t 2 I , where Y is any solution to (11). By Z we denote a measure on .I;ˇ.I // defined as Z .B/ WD Z .B ˝/ for B 2ˇ.I /: Obviously, Z D A C M . We will assume that semimartingale Z is such that Z is absolutely continuous with respect to the Lebesgue measure , and S Z WD ess sup The assertions concerning equation (11) will be stated under following assumptions: (Z3) there exists a continuous, nondecreasing, concave function ÄW OE0; 1/ ! OE0; 1/ satisfying: Ä.s/ ds D C1, Ä.0/ D 0, Ä.s/ > 0 for s > 0, and such that Z -a.e. it holds Due to concavity of Ä we obtain: , then there exists a constant K > 0 such that Z -a.e. it holds The existence of solution to (11) is established with a help of approximate solutions X n W I ! K b c .L 2 / (n D 1; 2; : : :) of Maruyama type, i.e.
for t 2 k 1 n ; k n i \ I , k D 1; 2; : : : For every n the mapping X n is H L 2 -continuous and X n .
then we can write F .s; Q X n .s//dZ.s/ for t 2 I: Proceeding similarly as in preceding section we obtain the following results. where Corollary 4.7. Let X 0 2 K b c .L 2 0 / and F satisfy (Z1)-(Z3). Then for n 2 N and t 2 I where M 2 is as in Lemma 4.6. To infer on continuous dependence of solution to (11) with respect to initial value, let us consider equation (11) and another one with different initial value . For the solutions X; Y of (11) and (12), respectively, the following estimation is true Corollary 4.11. Let X 0 2 K b c .L 2 0 / and F satisfy (Z1)-(Z3). Let X denote the solution of (11) and for n 2 N let X n denote the solution of equation H L 2 X n .t /; X.t / ! 0; as n ! 1: Now we consider two equations with two different nonlinearities F and V , i.e. equation (11) and . Let X denote solution to (11) and X n denote solution to the equation we can arrive to the following existence and uniqueness result.

Fuzzy stochastic equations driven by processes A, M from decomposition of semimartingale Z
In this section we present results parallel to those established for set-valued equations. However, here the equations will be more general. Namely, here the values of the integrands will be some fuzzy sets. Such equations are called the fuzzy stochastic integral equations. They can be viewed as equations in a metric space After setting an appropriate framework (see also [38,41] for more details), we formulate assertions for fuzzy stochastic equations driven by processes A; M from decomposition of Z. These results are generalizations of those included in Section 3. Their derivations are omitted, since they are similar to those presented in set-valued cases. Let F.X / denote the set of all functions uW X ! OE0; 1, where .X ; k k/ is a separable Banach space. Such set is a fundamental notion in fuzzy systems analysis (in industrial mathematics and engineering) and it is called the set of fuzzy sets of X . The function uW X ! OE0; 1 is called then a membership function of a fuzzy set u 2 F.X / and its value u.x/ is interpreted as a degree of membership of x in the fuzzy set u. Note that every ordinary subset A of X is a fuzzy set of X . Indeed, the power set of X can be embedded into F.X / by means of the characteristic function of ordinary set. Fuzzy sets play a key role in systems with imprecisely described states (not defined by a single value), in systems with incomplete information (e.g. with uncertain initial data) etc. Such systems often appear in realistic world. Therefore modeling uncertain systems with mathematical tools focuses much attention.
For˛2 .0; 1 the symbol OEu˛denotes the so-called˛-level set of u 2 F b c .X /, i.e.
OEu˛D fa 2 X W u.a/ >˛g; and OEu 0 will be used to denote supp.u/. It is clear that for u 2 F b c .X / we have OEu˛2 K b c .X / for every˛2 OE0; 1. Addition and scalar multiplication in F b c .X / is defined level-wise, i.e. the sum of u; v 2 F b c .X / is element u˚v 2 F b c .X / such that OEu˚v˛D OEu˛C OEv˛,˛2 OE0; 1, and the product of r 2 R and u 2 F b c .X / is an element ru 2 F b c .X / such that OEru˛D rOEu˛,˛2 OE0; 1. The set F b c .X / with these operations becomes a semilinear space.
The metric D X has the property An fuzzy stochastic process f W I ˝! F b c .X / is called: (a) predictable, if OEf . ; /˛W I ˝! K b c .X / is a predictable set-valued stochastic process for every˛2 OE0; 1, (b) L p P . A /-integrally bounded (L p P . M /-integrally bounded, L p P . Z /-integrally bounded), if the stochastic process .t; !/ 7 ! H X .OEf .t; !/ 0 ; fÂ X g/ belongs to L p .I ˝; P; A I R/ (.t; !/ 7 ! H X .OEf .t; !/ 0 ; fÂ X g/ belongs to L p .I ˝; P; M I R/, the process .t; !/ 7 ! H X .OEf .t; !/ 0 ; fÂ X g/ belongs to L p .I ˝; P; Z I R/, respectively). The element R t f .s/dA.s/ 2 F b c .L 2 t / from Proposition 5.1 is said to be the fuzzy stochastic trajectory integral (over interval OE ; t) of f with respect to the bounded variation process A, and R t g.s/dM.s/ 2 F b c .L 2 t / is said to be the fuzzy stochastic trajectory integral (over interval OE ; t ) of g with respect to the martingale M . Note that R t f .s/dA.s/˚R t g.s/dM.s/ 2 F b c .L 2 t /.
Also R t f .s/dA.s/; The following properties are counterparts of those listed in Proposition 2.3 and are needed in derivations of presented results concerning fuzzy stochastic equations.

Proposition 5.2 ([41]
). Let f 1 ; f 2 ; g 1 ; g 2 W R C ˝! F b c .R d / be the predictable fuzzy stochastic processes. Assume that processes f 1 ; f 2 are L 2 P . A /-integrally bounded, and g 1 ; g 2 are L 2 P . M /-integrally bounded. Then (i) for every ; a; t 2 I , 6 a 6 t  To obtain a result on existence and uniqueness of solution to (15) with the coefficients f; g satisfying the Osgood type condition we will assume that: (a1) the processes f . ; ; u/; g. ; ; u/W I ˝! F b c .R d / are predictable for every u 2 F b c .L 2 /, (a2) there exists a constant C > 0 such that A sequence fx n g 1 nD1 of approximate solutions x n W I ! F b c .L 2 / of Maruyama type is as follows: where W is fAg t -Wiener process we have shown [40] that the Euler-Maruyama approximations converge to the unique solution and this is achieved in [40] with the Lipschitz condition.
Note that each approximant x n is D L 2 -continuous and x n .t / 2 F b c .L 2 t / for every t 2 I , and if we define Proceeding similarly as in Section 3 with application of D L 2 and D R d instead of H L 2 and H R d , and applying Proposition 5.2 instead of Proposition 2.3, and applying Remark 5.4 instead of Remark 3.4, we get the following counterparts of results placed in Section 3. where Corollary 5.7. Assume that x 0 2 F b c .L 2 0 / and f; g satisfy (a1)-(a3). Then for n 2 N and t 2 I where M 2 is as in Lemma 5.6. The assertions written above are useful in derivation of existence of solution to (15).
with another initial value y 0 2 F b c .L 2 /.
Theorem 5.10. Assume that x 0 ; y 0 2 F b c .L 2 0 / and f; g satisfy (a1)-(a3). Then for the solutions x; y of (15) and (16), respectively, the following estimation is true Hence, continuous dependence of solution x to (15) with respect to initial value follows easily.
Corollary 5.11. Let x 0 2 F b c .L 2 0 / and f; g satisfy (a1)-(a3). Let x denote the solution of (15) and for n 2 N let x n denote the solution of equation Consider equation (15) and equation  Hence we can infer on continuous dependence of solution x with respect to the nonlinearities f; g.
. Let x denote solution to (15) and x n denote solution to the equation In this section we present results parallel to those presented in Section 4. First, we need to recall a notion of fuzzy stochastic trajectory integral.
Proposition 6.1 (cf. [38,42]). Assume that f W I ˝! F b c .R d / is a predictable and L 2 P . Z /-integrally bounded fuzzy stochastic process. Then for every ; t 2 I , < t there exists a unique element in F b c .L 2 t / denoted by   (iii) for every 2 I the mapping OE ; T 3 t 7 ! R t f 1 .s/dZ.s/ 2 F b c .L 2 / is continuous with respect to the metric D L 2 .
By a fuzzy stochastic integral equation driven by semimartingale Z we mean the following relation in the metric space F b c .L 2 /; D L 2 x.t / D x 0˚t Z 0 f .s; x.s//dZ.s/ for t 2 I; where f W I ˝ F b c .L 2 / ! F b c .R d / and x 0 2 F b c .L 2 0 /. The definitions of solution to (18) and its uniqueness are similar to those presented in Definition 5.3. To obtain a result on existence and uniqueness of solution to (18) we will assume that: (z1) f . ; ; u/W I ˝! F b c .R d / is predictable for every u 2 F b c .L 2 /, (z2) there exists a constant C > 0 such that Z -a.e. it holds with another initial value y 0 2 F b c .L 2 /.

Concluding remarks
In this paper, we study the set-valued and fuzzy stochastic integral equations. These equations generalize the classical single-valued stochastic differential equations [47,48], random differential equations [54], deterministic set-valued and fuzzy differential equations [31,32]. Our analysis concerns the equations driven by semimartingales that constitute the largest class of integrators with respect to which stochastic integrals can be reasonably defined. Up until now, the investigations have been undertaken in a framework of the Lipschitzian coefficients of the equations [38][39][40][41][42]. In this paper we relax the assumptions imposed on coefficients and require only that they satisfy an Osgood type condition, which is weaker than the Lipschitz one [46]. In this way, a class of admissible integrands is extended. The fixed point theorems and Gronwall's lemma [38,39,41,42] are the main tools in derivations concerning the set-valued and fuzzy stochastic integral equations under the Lipschitz condition. For the equations with coefficients satisfying Osgood's type condition we use Maruyama's successive approximations method and the Bihari's inequality. We show that the set-valued and fuzzy Maruyama's approximations sequences are in fact certain Cauchy sequences with their limits being the desirable solutions. The uniqueness of the solution is showed with a help of Bihari's inequality, which is also applied in proving the continuous dependence of solutions with respect to initial value and nonlinearities of the equation.
The results established for the set-valued and fuzzy stochastic integral equations have immediate implications for the single-valued stochastic integral equations driven by semimaringales. The latter are studied in [48] under the Lipschitz condition.