Properties of Solutions to Stochastic Set Differential Equations under Non-Lipschitzian Coefficients

and Applied Analysis 3


Introduction
Set-valued differential equations which were started in 1969 by de Blasi and Lervolino [1] have been employed in investigations of dynamic systems.The evidence of set differential equations for such areas as control theory, differential inclusions, and fuzzy differential equations can be found in [2][3][4][5][6][7][8][9][10][11][12] and references therein.The set differential equations also are explored in [13][14][15].One of the main advantages of investigating deterministic set differential equations is that they can be used as a tool for studying properties of solutions of differential inclusions.On the other hand, the set-valued random processes are first introduced by van Cutsem [16].Since then the subject has attracted the interest of many mathematicians and further contributions are made from both the theoretical and applied viewpoints (see, e.g., [17][18][19][20][21][22][23][24][25][26]).In [27][28][29][30][31], the set-valued random differential equations are studied.The strong solution of Itô type set-valued stochastic differential equation is analyzed in [32].
As far as we know, there exists a wide literature where attempts have been made to investigate stochastic differential inclusions (see, e.g., [33][34][35][36][37][38][39][40] and references therein).And recently, in [27], a kind of the SSDEs disturbed by Wiener processes is investigated, where under the Lipschitzian condition the existence and uniqueness of solutions to the SSDEs are proven.Under the non-Lipschitzian condition, the existence and uniqueness of solutions to the stochastic set differential equations are proven in [41,42].Moreover, in our present paper, under the non-Lipschitzian condition the nonexplosion and continuous dependence of solutions to the SSDEs are studied.The mathematical tool employed in the paper is the Bihari inequality and the notion of the support function.The work presented here generalizes results obtained both for deterministic and for random set differential equations.Also, it should be noted that the work related to this paper is the discussions of fuzzy-valued processes and stochastic differential equations (see, e.g., [43][44][45][46][47][48][49][50][51]).
The paper is organized as follows.Section 2 gives an appropriate framework on a set-valued analysis within which the notion of a set-valued stochastic integral is given.In Section 3, moreover, the continuous dependence of the solutions for SSDEs on initial conditions and nonexplosion are discussed.Finally, the conclusions are made in Section 4.

Preliminaries
Let K(R  ) be the family of all nonempty compact and convex subsets of R  .In K(R  ), we define the Hausdorff metric   of two sets ,  ∈ K(R  ) as follows: Throughout this paper, let (Ω, A, ) be complete probability space.
Abstract and Applied Analysis multifunctions with values in R  .A multifunction  ∈ M(Ω, A; K(R  )) is said to be   -integrably bounded,  ≥ 1, if there exists ℎ ∈   (Ω, A, ; R + ) such that |‖‖| ≤ ℎ a.s., where Let us denote Denote  := [0, ∞).Let (Ω, A, {A  } ∈ , ) be a complete, filtered probability space where the sub--field family (A  ,  ∈ ) of A satisfies the usual conditions.We call  :  × Ω → K(R  ) a set-valued stochastic process, if for every is {A  } ∈ -adapted and measurable, then it will be called nonanticipating.Equivalently, the set-valued process  is nonanticipating if and only if  is measurable with respect to the -algebra N, which is defined as follows: where   = { : (, ) ∈ } for  ∈ .
We say that a set-valued stochastic process  is  continuous, if almost all its trajectories, that is, the mappings In what follows, we state the generalized Bihari inequality (cf., Mao [52]) which plays an important role in the following section.
(i) If () is a continuous nonnegative nondecreasing function on [0, ], then the inequality implies that where where  −1 is the inverse function of  and  ∈ [0, ].

Properties of Solutions to SSDEs
In this section, we consider the following stochastic set differential equation (in the integral form): where  takes values in Brownian motion, and stands for the family of the continuous functions from the space K(R  ) to the space In (16), the integral ∫  0 (()) is Aumann's one, and the integral ∫  0 (())() is a general Itô type stochastic one, whose definition can refer to [53].
It is known that (17) has a solution up to a lifetime () which depends on  ∈ B. Set We call  the lifetime of solution to (16).Obviously, ‖|()|‖ = +∞ a.s.Here, by the concepts of explosion time and lifetime time from Pages 158 and 191 in [54], the lifetime of solution to (16) or (17) is the same as the explosion time of the solution of ( 16) or (17).If (17) has the pathwise uniqueness, then we show the existence and uniqueness of the solution to the SSDE (16).So the study of pathwise uniqueness is of great interest.It is a classical result that, under the Lipschitz coefficients, the pathwise uniqueness holds and the solution of ( 17) can be constructed by using Picard iteration; moreover, the solution depends on the initial values continuously.However, under non-Lipschitzian condition, Fei [42] presents the existence and uniqueness of solutions to (16); hence, the existence and uniqueness of solutions to (17) also are proven.
In what follows, we discuss the nonexplosion of the solutions to ( 16) under non-Lipschitzian condition.Our idea is to derive an inequality so that the generalized Bihari inequality (Lemma 1) can be applied.
where we have utilized the concavity of the functions V(V) and √V.
Proof.By constructing the sequence {  } of set-valued random variables as that in [42], the existence of the solutions to ( 16) is similarly proven.Next, we prove the uniqueness of the solutions to (16).Let (()) ≥0 and (()) ≥0 be two solutions of (16).Set which deduces where we have utilized the concavity of the functions V(V).
By Lemma 1 (i), we obtain which shows that () = ().Thus we complete the proof.
Note that function is a typical example satisfying conditions (i) and (ii).
Next, we will study the dependence of the solutions to the SSDE ( 16) on initial data.For the mapping  0 → (,  0 ), we call   ( 0 ) = (,  0 ) mean square continuous on  0 , uniformly with respect to  ∈  if   ((,  0 ), (,  0 )) → 0 as   − lim  0 =  0 on any compact subset  of , where the limit  0 of  0 is in sense of the metric   .Theorem 4. Assume that the conditions in Theorem 3 hold.Then the mapping  0 →   ( 0 ) is mean square continuous, uniformly with respect to  in any compact subset, where   ( 0 ) = (,  0 ) is the solution to SSDE (16).
By Lemma 1 (i), we have For arbitrary  > 0 and given , it is easy to deduce that there exists  > 0 with   ( 0 ,  0 ) < , which shows   ( 0 ,  0 ) < , such that Since () is increasing, we have that Thus, we show   ( 0 ) is mean square continuous on  0 , with respect to  in any compact subset.Therefore, we complete the proof.
The following theorem gives the sufficient condition.

Conclusions
In many real dynamic systems, we are often faced with random experiments whose outcomes might be multivalued.Moreover, the stochastic set differential equations may be employed in characterizing a large class of physically important dynamic systems which can be applied in such areas as control, economics, and finance.In this paper, we study the behavior of solutions to SSDEs disturbed by a Wiener process with the non-Lipschitzian coefficients.First, the nonexplosion theorem of the Itô type SSDEs is proven.Then the existence and uniqueness theorem of solutions to SSDEs is given.Moreover, the continuous dependence of solutions to the SSDEs is investigated.Main mathematical tool is the notion of the support function and the generalized Bihari inequality.Besides, the present case can be extended to the SSDEs driven by a multidimensional semimartingale in future.