Lebesgues-Stieltjes Integrals of Fuzzy Stochastic Processes with Respect to Finite Variation Processes

Let ( ) [ ] { } t G G t T = ∈ ω , 0, be a fuzzy stochastic process and ( ) [ ] { } t A t T ∈ ω , 0, be a real valued finite variation process. We define the Lebesgue-Stieltjes integral denoted by ( ) ( ) ∫ t s s G A ω ω 0 d for each t > 0 by using the selection method, which is direct, nature and different from the indirect definition appearing in some references. We shall show that this kind of integral is also measurable, continuous in time t and bounded a.s. under the Hausdorff metric.


Introduction
Recently, the theory of fuzzy functions has been developed quickly due to the measurements of various uncertainties arising not only from the randomness but also from the vagueness in some situations.For example, when considering wave height at time t denoted by t f , due to the influence of random factors and the limitations of the measurement tools and methods, we may not precisely know the height t f .It is reasonable to consider the wave height as a fuzzy random variable on a probability space ( ) , ,P Ω  .Since Puri and Ralescu [1] (1986) defined fuzzy random variable, there had been many further topics such as expectations of fuzzy random variables, fuzzy stochastic processes, integrals of fuzzy stochastic processes, fuzzy stochastic differential equations etc.In order to study a fuzzy function u, it is natural and equivalent to study its α -level set [ ] , where [ ] u α is a set-valued function.Therefore, as usual, in order to explore the integrals of fuzzy stochastic processes, at first we can study the integrals of set-valued stochastic processes.Kisielewicz (1997) [2] used all selections to define the integral of a set-valued process as a nonempty closed subset of ( ) 2 , , ; n L P R Ω  , but did not consider its measurability.Based on Kisielewicz's work (1997) [2], Kim and Kim (1999) [3] studied some properties of this kind of integral.Jung and Kim (2003) [4] modified the definition in 1-dimensional Euclidean space R so that the integral became a set-valued random variable.After the work [4], there are some references on set-valued integrals and fuzzy integrals.One may refer to papers such as [5]- [13] etc. and references therein.Zhang and Qi [14] (2013) considered the set-valued integral with respect to a finite variation process directly instead of taking the decomposable closure appearing in [4] [6] and other references.As a further work of [14], here we shall explore the integrals of fuzzy stochastic processes with respect to finite variation processes and prove the measurability and boundedness of this kind of integral, the continuity with respect to t under the Hausdorff metric and its representation theorem.
This paper is organized as follows: in Section 2, we present some notions on set-valued random variables and fuzzy set-valued random variables; in Section 3, we shall give the definition of integral of fuzzy set-valued stochastic processes with respect to finite variation process and prove the measurability and 2  L -boundedness which are necessary to our future work on fuzzy stochastic differential equations.

Preliminaries
We denote N the set of all natural numbers, R the set of all real numbers, d R the d-dimensional Euclidean space with the usual norm ⋅ , R + the set of all nonnegative numbers.Let ( ) a σ -field filtration satisfying the usual conditions such that 0  includes all P-null sets in  .The filtration is non-decreasing and right continuous.Let ( ) be a Borel field of a topological space E.

Let ( )
, ,P Ω  be a complete probability space.

( )
, , ; ( ) ) be the family of all nonempty, closed (resp.nonempty compact, nonempty compact convex) subsets of d R .For any , define the distance between x and A by ( ) [15]) is defined by , the support function of A is defined as follows: , , ; , , ; , , ; ) be the family of all measurable ( ) , , ; , , ; , the family of all p L -integrable se- lections is defined by In the following, ( )

( )
x F ω ∈ , ( ) ) is defined as a func- tion : with the  -measurable section t f , for 0 t ≥ .We say f is measurable if f is ( ) is measurable and t  -adapted if and only if it is Σ -measurable ( [9]).
In a fashion similar to the d R -valued stochastic processes, a set-valued stochastic process which satisfy the following two conditions (cf.[3] [6]): 1) The level set denote the family of all fuzzy sets which satisfy the above conditions 1), 2), and  ( ) be the subset of all convex fuzzy sets in ( ) [1]) by the expression ). Moreo- ver, for every ( ) , , , , [16]) Let B be a set and Then the function . Such a mapping G is called a fuzzy random variable (cf.[17]).Let ( ) ( ) , , ; ) denote the family of all  -measurable fuzzy random variables.As a similar manner, we have the notations , , ; , , ; , : A fuzzy stochastic process G is called p L -integrably bounded, if there exists a real-valued stochastic process , , ;  ( ) , where we consider be a fuzzy random variable and 1 p ≥ , The following conditions are equivalent (cf.[15]): , , ; We define ( ) , where for

Lebesgue-Stieltjes Integrals with Respect to Finite Variation Processes
Let ( ) , ,P Ω  be a complete probability space equipped with the usual filtration 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 [ ] [ ) , 0, s t ⊂ ∞ and any partition is the decomposition of A. A + and A − are non-negative and non-de- creasing processes.[7] defined a measure as follows:   [7]).In the following we always assume Lemma 2. (cf.[8]) Let ( ) , , E µ  be a σ -finite measure space and X a separable Banach space.If  is separable with respect to µ , (i.e.there exists a countably generated sub-sigma algebra 0 ∈   such that for every A ∈  , there is is separable in norm.From now on, we always assume the sigma-field  is separable with respect to P such that the set-valued integral and fuzzy integral can be well defined. Let For any [ ] ( ) ( ) 0, , , ; .

with respect to the finite variation continuous process A is the set
.
For some fuzzy stochastic process , it is natural to define the fuzzy integral of G with respect to the finite variation process level-wise.

M T F R Ω×
) be the family of all Σ -measu- rable ( ) .
For a fuzzy stochastic process , according to Lemma 1 and the properties of set-valued stochastic integrals, the Lebesgue-Stieltjes integral of G (over interval [ ] , s t ) can be defined level-wise. Set and ω ∈ Ω , the Lebesgue- Stieltjes integral ( ) ( ) ∫ is a compact and convex subset of d R .

for each x X
∈ , the function ( ) is  -measurable and for each ω ∈ Ω , the function ( ) , the fuzzy stochastic integral is Σ -measurable too.By Lemma 4, we have ∫ is a Carathedory function, then by Lemma 6, we obtain that We will show that for any which means ∫ is continuous with respect to t under the metric d ∞ .
Proof.Let 0 r t T ≤ < ≤ , for any ω ∈ Ω , we have For any ω ∈ Ω , we have . Therefore it is continuous in t with respect to d ∞ .
Lemma 7. Let fuzzy stochastic process where the closure is taken in Since  is separable with respect to probability measurable P, we have that [ ] ( ) Theorem 5.For a fuzzy set-valued stochastic process and for each t . .
where "cl" denotes the closure in d R .
, by Lemma 7, there exists a sequence { } [ ] ( ) where the closure is taken in , by Castaing represent theorem (cf.[15] [20]), there exists a sequence At first we will show that (

In fact, taking
[ ] , there exists a sequence { } then there exists a subsequence { } On the other hand , which yields , then for each t .
φ ω is measurable with respect to ω , then there exists .
In Theorem 6 and Theorem 7, the inequalities hold too if we take the expectation on both sides.

Conclusion
In [21], the author studied the Lebesgue-Stieltjes integral of real stochastic processes with respect to fuzzy valued stochastic processes.In some references such as [5] [6], the integrals of fuzzy stochastic processes with respect to time t and Brownian motion were studied.In order to guarantee measurability of the integral, Kim (2005) Li and Ren (2007) defined the integral indirectly by taking the decomposable closure.Here, when the integrand taked value in compact and convex subsets of ( ) d F R , we defined directly the integral of fuzzy sto- chastic process with respect to real-valued finite variation processes by using selection method, which is different from the above references.Then we proved the measurability (Theorem 2), which was key and guaranteed the reasonability of the definition.Attribute to the good property of finite variation of integrator, the integral was bounded as and 2  L -bounded under the metric d ∞ (Theorem 3, Theorem 6 and Theorem 7).This property was much well than the integral with respect to Brownian motion since the latter was of infinite variation.Thanks to the boundedness of the integral, it was possible to do the further work such as exploring solutions of fuzzy stochastic differential equations. d

pL
-integrably bounded fuzzy functions.Write for brevity by ( ) where C I is the index function.Then the set function v is a finite measure in the measurable space

∫,
defined by Equation (3) can determine an such a fuzzy function is called the Lebesgue-Stieltjes integral (over interval [ ], s t ) of G with respect to finite variation process( ) .