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Let be a fuzzy stochastic process and be a real valued finite variation process. We define the Lebesgue-Stieltjes integral denoted by for each 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.

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

Since Puri and Ralescu [

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

We denote N the set of all natural numbers, R the set of all real numbers,

Let

is finite. f is called

Let

Denote

For

A set-valued function

Let

In the following,

A set-valued random variable F is said to be integrable if

An

In a fashion similar to the

Let

1) The level set

2) Each v is upper semi-continuous function, i.e. for each

Let

3) The support set

A fuzzy set v is convex if

It is know that v is convex if and only if, for any

sets in

We know that

Lemma 1. (cf. [

1)

2)

3)

Then the function

A mapping

A fuzzy stochastic process G is called

Let

Let

1)

2)

3)

We define

Let

is finite and

where

In the product space

For

For

For any

Lemma 2. (cf. [

From now on, we always assume the sigma-field

Let

where

For any

Definition 1. (cf. [

For some fuzzy stochastic process

Let

where

For a fuzzy stochastic process

Set

for all

Definition 2. For a fuzzy stochastic process

by

Theorem 1. ([

Lemma 3. (cf. [

Lemma 4. (cf. [

1)

2)

Lemma 5. (cf. [

1)

2) for any

We obtain that F is a set-valued random variable.

From Lemma 3 and Lemma 5, when

Lemma 6. (cf. [

Theorem 2. Let

Proof. Taking

Theorem 3. Let

Proof. By Theorem 2, for any

For any

Then

Hence,

which means

Theorem 4. Let

Proof. Let

Then

For any

Then for all

Lemma 7. Let fuzzy stochastic process

where the closure is taken in

Proof. Since

Theorem 5. For a fuzzy set-valued stochastic process

and for each t

where “cl” denotes the closure in

Proof. For each

where the closure is taken in

For each

At first we will show that

In fact, taking

then there exists a subsequence

Therefore

On the other hand

since

Since

is closed and

For any

Then for each t,

which means

(7) together with (8) yields

Lemma 8. (cf. [

Theorem 6. Let

Proof. Let

and

Therefore

By Lemma 8, we have

Then

Then

Similarly, we have

Then for each

Therefore

Hence

Theorem 7. Let

Proof. For any

by Lemma 8, we have

Then

Then

Similarly, we have

Then for each

Moreover

Hence

Remark 1. In Theorem 6 and Theorem 7, the inequalities hold too if we take the expectation on both sides.

In [

We thank the editor and the referees for their comments. This work is partly supported by NSFC (No. 11371135).

JinpingZhang,LingliLuo,XingmeiLi,XiaoyingWang, (2015) Lebesgues-Stieltjes Integrals of Fuzzy Stochastic Processes with Respect to Finite Variation Processes. Applied Mathematics,06,2199-2210. doi: 10.4236/am.2015.613193