Set-valued Stochastic Integrals with Respect to Finite Variation Processes

In a Euclidean space , the Lebesgue-Stieltjes integral of set-valued stochastic processes d R       , 0, t F F t T    with respect to real valued finite variation process       , 0, t A t T   t is defined directly by employing all integrably bounded selections instead of taking the decomposable closure appearing in some existed references. We shall show that this kind of integral is measurable, continuous in under the Hausdorff metric and-bounded.


Introduction
Recently, integrals for set-valued stochastic processes with respect to Brownian motion, martingales and the Lebesgue measure have received much attention.
In 1997, Kisielewicz ([1]) defined the integral of setvalued process as a subset of space, but he didn't consider the measurability of the integral.In 1999, Kim and Kim [2] used the definition of stochastic integrals of set-valued stochastic process with respect to the Brownian motion.They called it Aumann ( [3]) type It integrals.In [4], Jung and Kim modified the definition by taking the decomposable closure such that the integral is measurable.Li and Ren [5] modified Jung and Kim's definition by considering the predictable set-valued stochastic process as a set-valued random variable in the product space , and the measurability and decomposability also were based on product After that, Zhang et al. ([6,7]) studied the set-valued integrals with respect to the martingale and Brownian motion.
Stochastic differential inclusions and set-valued stochastic differential (or integral) equations are employed to model the problems with not only randomness but also impreciseness.Recently, there are some references related to set-valued differential equations such as [8][9][10][11][12][13] etc.
Concerning to the integral with respect to finite variation processes, Malinowski and Michta [12] give the notion of set-valued integral with respect to single valued finite variation but without considering the measurability.Z.Wang and R.Wang [14] defined the Lebesgue-Stieltjes stochastic integral of single valued stochastic processes with respect to set-valued finite variation processes (refer to [14] for the detail).
In this paper, different from the definition in [14], based on the Definition 3.1 in [12], we will study the Lebesgue-Stieltjes integral of set-valued stochastic processes with respect to single valued finite variation process.We shall prove the measurability of integral, namely, it is a set-valued random, which is similar to the classical stochastic integral.
This paper is organized as follows: in Section 2, we present some notions and facts on set-valued random variables; in Section 3, we shall give the definition of integral of set-valued stochastic processes with respect to finite variation process and then prove the measurability and -boundedness.

Preliminaries
We denote the set of all natural numbers, the set  R of all real numbers, the d-dimensional Euclidean space with the usual norm d R  , the set of all nonnegative numbers.Let  be a complete probability space,  -field filtration sa- tisfying the usual conditions.Let be a Borel field of a topological space .
  ) be the family of all nonempty, closed (resp.nonempty compact, nonempty compact convex) subsets of .For any , define the distance between x and A by   (see e.g.[15]) is defined by the support function of A is defined as follows: is said to be measurable if for any open set , the inverse belongs to  .Such a function ) be the family of all measurable -valued (resp.
-valued) functions, briefly by (resp. .For , the family of all   , , ; , , ; In the following, is denoted briefly by is measurable and -adapted if and only if it is -measurable ( [8]). t In a fashion similar to the -valued stochastic processes, a set-valued stochastic process -measurable, and tadapted if for any fixed t , is t -measurable.

Set-Valued Stochastic Integral w.r.t Finite Variation Processes
be a real valued t -adapted measurable process with finite variation and continuous sample trajectories a.s.from the origin.That is to say, for each compact interval and any partition is finite and   0, 0 A   a.s.Then for any T , the process can generate a random measure denoted by A  in the space .For any is the decomposition of A , A  and A  are non-negative and nondecreasing processes, for   , where is the index function.Then the set function In the following we always assume Note: in [12], the i being predictable, in fact the integ can to the ntegrand is assumed rand be relaxed : 0 , , T , ; : , , , ..
Definition 1. (see [12]) For a set-valued stochasti process the set-valued st c ) of F with respect to the finite variation continuous process A is the set d : In [12], the authors call this kind of integral as tra ctory integral since they consider it as a and show nt je -valued random variable.Here, we shall consider it as a subset of d  the measurability with res R pect to  , which is very different from the way in [12], also differe from other references such as [10,16,17] , therefore for every set it is well defined tinuity of since the con t A .In the sequel, we shall denote the integral by ve.The nvexity of the integral is obvious.
ar operator mplies the linear operator is bounded.ore the integral is weakly compact since y 2

the bounded linear operator mapping akly compact set to a weakly compa
In d R space, a weakly compact set is compact.
By using Lemma 1, as eorem 1 in [17], we a manner similar to Th have the following result: Lemma 2. Assume A is the corresponding stochastic process, , we h e  n The F is a set-valued random variable.a 1 and Lemma 3, wh , for any From Lemm en ly measurable.Then by Lemma 1 we have the following: By Theorem 1, we have for all    .Furthermore, we obtain for all Moreover, since measurable, from the Lemma 5 we can obtain th , in is Finally argument above, the function , , we will show th As a manner similar to Theorem 3.8. in [8] the Castaing representation as following: Theorem 3.For a set-valued stochastic process , we have   where cl denotes the closure in Theorem 4. For each   0, ; , , is continuo us a.s. with respect to the Haus- Hence, etc.In fact, for almost every    , the above integral     F   1 F  and for every , such that  P Vol. , pp. 480-502.http://dx.doi.org/10.1006/jmaa.1999.6461