Set-Valued Stochastic Equation with Set-Valued Square Integrable Martingale

In this paper, we shall introduce the stochastic integral of a stochastic process with respect to set-valued square integrable martingale. Then we shall give the Aumann integral measurable theorem, and give the set-valued stochastic Lebesgue integral and set-valued square integrable martingale integral equation. The existence and uniqueness of solution to set-valued stochastic integral equation are proved. The discussion will be useful in optimal control and mathematical finance in psychological factors.


Introduction
Set-valued theory is used in optimal control(cf.[1]), mathematical finance(cf.[2]), fixed point theory(cf.[3]).Set-valued and fuzzy set-valued theory can be used to account for psychological factors(cf.[4,5]).In [6], stochastic control problems are discussed by stochastic integral with respect to set-valued square integral martingales.M. Malinowski et al. discussed the setvalued stochastic integral driven by semi-martingale and the set-valued stochastic differential equations driven by semi-martingale in [7].J. Li et al. discussed setvalued stochastic Lebesgue integral and set-valued stochastic differential equaton in [8,9,10].Puri and Ralescu defined fuzzy set-valued martingales and proved convergence theorems of fuzzy set-valued martingales in [11].J. Li et al. discussed the space of fuzzy set-valued square integrable martingales in [12] and fuzzy set-valued stochastic Lebesgue integral in [13].
In this paper, we shall give the set-valued stochastic integral equation : the first integral is set-valued stochastic Lebesgue integral (see [9,10]), the second integral is set-valued square integrable martingale integral (see [6]).Aumann integral measurable theorem of the second integral shall be given.The existence and uniqueness of the solution to the equation shall be proved.
We organize our paper as follows: in Section 2, we shall introduce some necessary notations, definitions, and results about set-valued stochastic variables and martingales.
Furthermore, we shall give the Aumann integral measurable theorem.In Section 3, we shall give the set-valued stochastic integral equation with respect to set-valued square integrable martingale, and prove the existence and uniqueness of the solution to the set-valued stochastic integral equation.

Preliminary on Set-Valued Random Variables and Martingales
Throughout this paper, assume that ( , : ࣛ, ) P is a complete probability space, the V -field filtration { ࣛ t : } t I satisfies the usual conditions (i.e.Containing all nullsets, non-decreasing and right continuous), [0, ] I T with 0 T > , R is the set of all real numbers, N is the set of all natural numbers, The Hausdorff metric on ( ) satisfies that for any open set , then F is called ࣛ -measurable (or a set-valued random variable, random set, multivalued function.).Let { ( ), Î is adapted and for any a.e.

( P ) .
A set-valued martingale { ( ), Denote ( ) CMS F as the set of d R -valued continuous martingale selections of set-valued square integrable martingale F .
Concerning more notations, definitions and more results of set-valued random variable, set-valued martingales, readers could refer to [6,14].
Definition (cf.[6]) Assume { ( ),  The above theorem is useful to discuss the theory of Aumann integral, etc. respect to set-valued square integral martingale (see Definition 4.5 in [6]).
The uniqueness of the solution is the same to the proof of the existence.

Remark 3
Furthermore, fuzzy set-valued martingale integral, fuzzy set-valued stochastic Lebesgue integral [13], and fuzzy set-valued stochastic integral would be useful in the area.

d
dimensional Euclidean space with usual norm || || × , ी ( ) E is the Borel field of the space E .Let ( ) d K R be the family of all nonempty, closed subsets of d R .nonempty subsets of d R , define the distance between x and A , -valued random variable.

a
> is a constant;(3) Set-valued integral inequality: it is obviously right.Suppose that n F has properties ( ),( ) a b for any fixed n , we shall prove so By the Linear bound condition (1) and norm property of set-valued Lebesgue integral, it satisfiesLetIt is obviously that ( ) a holds.By Theorem 1 and Theorem 4.7 in[6], the set-

Step 2 .
Now we shall prove that n F convergences to F .Let We have For the first part, we have For the second part, by inequality (3), we have DOI: 10.1051/ , 03002 way, we have Iterating the above process, we get Thus, Therefore, we have This ensures the existence of the strong solution.
[6]ann type stochastic integral of g with respect to the set-valued square integrable martingale F .By Therom 4.4 and Therom 4.7 in[6], we have the following Aumann integral measurable theorem: I Î is a separable square integrable set-valued martingale and for any fixed , ( , ) F t