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

Abstract

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.

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Zhang, J. , Luo, L. , Li, X. and Wang, X. (2015) Lebesgues-Stieltjes Integrals of Fuzzy Stochastic Processes with Respect to Finite Variation Processes. Applied Mathematics, 6, 2199-2210. doi: 10.4236/am.2015.613193.

Received 28 September 2015; accepted 27 November 2015; published 30 November 2015  1. 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 , due to the influence of random factors and the limitations of the measurement tools and methods, we may not precisely know the height . It is reasonable to consider the wave height as a fuzzy random variable on a probability space .

Since Puri and Ralescu  (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 -level set for any , where 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)  used all selections to define the integral of a set-valued process as a nonempty closed subset of , but did not consider its measurability. Based on Kisielewicz’s work (1997)  , Kim and Kim (1999)  studied some properties of this kind of integral. Jung and Kim (2003)  modified the definition in 1-dimensional Euclidean space R so that the integral became a set-valued random variable. After the work  , there are some references on set-valued integrals and fuzzy integrals. One may refer to papers such as  - etc. and references therein. Zhang and Qi  (2013) considered the set-valued integral with respect to a finite variation process directly instead of taking the decomposable closure appearing in   and other references. As a further work of  , 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 -boundedness which are necessary to our future work on fuzzy stochastic differential equations.

2. Preliminaries

We denote N the set of all natural numbers, R the set of all real numbers, the d-dimensional Euclidean space with the usual norm , the set of all nonnegative numbers. Let be a complete probability space, a -field filtration satisfying the usual conditions such that 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 be a complete probability space. (or brief) the set of all -valued Borel measurable functions such that the norm

is finite. f is called -integrable if.

Let (resp.,) be the family of all nonempty, closed (resp. nonempty compact, nonempty compact convex) subsets of. For any and, define the distance between x and A by. The Hausdorff metric on (cf.  ) is defined by

Denote

For, the support function of A is defined as follows:

A set-valued function is said to be measurable if for any open set, the inverse belongs to. Such a function F is called a set-valued random variable.

Let (resp.),) be the family of all measurable -valued (resp., -valued) functions, briefly by (resp.,. For, the family of all -integrable selections is defined by

(1)

In the following, is denoted briefly by.

A set-valued random variable F is said to be integrable if is nonempty. F is called -integrably bounded if there exits s.t. for all, almost surely.

An -valued stochastic process (or denoted by) is defined as a function with the -measurable section, for. We say f is measurable if f is -measurable. The process is called -adapted if is -measurable for every. Let, where. We know that is a -algebra on. A function is measurable and -adapted if and only if it is -measurable ( ).

In a fashion similar to the -valued stochastic processes, a set-valued stochastic process is defined as a set-valued function with -measurable section for. It is called measurable if it is -measurable, and -adapted if for any fixed t, is -measurable. is measurable and -adapted if and only if it is -measurable. is called -integrable if every is -integrable.

Let denote the family of all fuzzy sets which satisfy the following two conditions (cf.   ):

1) The level set;

2) Each v is upper semi-continuous function, i.e. for each, the level set is a closed subset of.

Let denote the family of all fuzzy sets which satisfy the above conditions 1), 2), and

3) The support set is a compact set.

A fuzzy set v is convex if

It is know that v is convex if and only if, for any, the level set is a convex subset of. Let denote the family of all convex fuzzy sets in, and be the subset of all convex fuzzy

sets in. Define (cf.  ) by the expression

We know that is a metric in and a complete metric space (cf.   ). Moreover, for every, we have

Lemma 1. (cf.  ) Let B be a set and be a family of subsets of B such that

1);

2) implies;

3), implies.

Then the function defined by has the property that for every.

A mapping is said to be measurable if is an set-valued random variable for each. Such a mapping G is called a fuzzy random variable (cf.  ). Let (briefly by) denote the family of all -measurable fuzzy random variables. As a similar manner, we have the notations, and, or briefly by (resp.).

is called a fuzzy stochastic process if for any, is a fuzzy random variable. A fuzzy stochastic process is said to be -adapted, if for every, the set-valued function is -measurable for all. It is called measurable, if is a -measurable for all.

A fuzzy stochastic process G is called -integrably bounded, if there exists a real-valued stochastic process, for any such that for any. It is equivalent to that.

Let denote the family of all measurable -valued -integrably bounded fuzzy functions. Write for brevity by, where we consider as identical if. Let denote the family of all -integrably bounded -valued -adapted fuzzy stochastic processes.

Let be a fuzzy random variable and, The following conditions are equivalent (cf.  ):

1);

2);

3).

We define as, where for, we have if and if.

3. Lebesgue-Stieltjes Integrals with Respect to Finite Variation Processes

Let be a complete probability space equipped with the usual filtration. Let be a real valued -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 of, the total variation

is finite and as. Then for any, the process can generate a random measure denoted by in the space. For any, let

where is the decomposition of A. and are non-negative and non-de- creasing processes..

In the product space, Michta (2011) in  defined a measure as follows:

For, where is the index function. Then the set function v is a finite measure in the measurable space if and only if (cf.  ). In the following we always assume .

For, let be the family of all -measurable -valued stochastic processes f such that

For any and, the stochastic Lebesgue-Stieltjes integral is defined by the Bochner integral path-by-path. One can prove that the integral process is -measurable.

Lemma 2. (cf.  ) Let be a -finite measure space and X a separable Banach space. If is separable with respect to, (i.e. there exists a countably generated sub-sigma algebra such that for every, there is satisfying), then space 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 be the family of all -measurable -valued stochastic processes F such that

where.

For any, set

(2)

Definition 1. (cf.  ) For a set-valued stochastic process the set-valued stochastic Lebesgue-Stieltjes integral (over interval) of F 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.

Let (or abbrev. as) be the family of all -measu- rable -valued fuzzy stochastic processes G such that

where.

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) can be defined level-wise.

Set

(3)

for all.

Definition 2. For a fuzzy stochastic process and any, the family

defined by Equation (3) can determine an -valued function denoted

by, such a fuzzy function is called the Lebesgue-Stieltjes integral (over interval) of G with respect to finite variation process.

Theorem 1. ( ) For, and, the Lebesgue- Stieltjes integral is a compact and convex subset of.

Lemma 3. (cf.  ) Let be a probability space, X a separable Banach space. For random variables, both the support function and the metric are -measurable.

Lemma 4. (cf.  ) Let be an R-valued stochastic process with finite variation. For and, we have

1);

2).

Lemma 5. (cf.  ) Let be a measurable space, X a separable Banach space. Taking and for any, assume is measurable. Then if one of the following conditions is satisfied:

1) is separable;

2) for any.

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

From Lemma 3 and Lemma 5, when, taking, then for any, is measurable if and only if is -measurable.

Lemma 6. (cf.  ) Let be a measurable space, X a separable metrizable space, and Y a metrizable space. Then every Caratheodory function ( i.e. for each, the function is -measurable and for each, the function is continuous) is -mea- surable.

Theorem 2. Let. Then for each, the fuzzy stochastic integral is -measurable. Furthermore, the mapping is -measurable.

Proof. Taking, then for each, the mapping is -measurable. For any, by Lemma 3, the support function is -measurable too. By Lemma 4, we have . Since the real-valued Lebesgue-Stieltjes integral is a Carathedory function, then by Lemma 6, we obtain that is -measurable. Therefore, by Lemma 5, for each, the mapping is -measurable and -adapted, which means the integral is -measurable and -adapted.

Theorem 3. Let. Then for any,.

Proof. By Theorem 2, for any, is -measurable. We will show that for any, ,.

For any,

. (4)

Then

Hence,

(5)

which means.

Theorem 4. Let. Then for any, is continuous with respect to t under the metric.

Proof. Let, for any, we have

Then

(6)

For any, we have

Then for all, is left continuous for under the metric. Similarly, we can prove that is a right continuous for. Therefore it is continuous in t with respect to.

Lemma 7. Let fuzzy stochastic process. Then for each, there exists a sequence, such that for every,

where the closure is taken in.

Proof. Since is separable with respect to probability measurable P, we have that is separable with respect to product measure. By Lemma 2, is separable. It can be obtained that is separable under the norm. So that for any, is separable since it is a closed subset of. Then there exists a sequence ,

Theorem 5. For a fuzzy set-valued stochastic process and any, there exists a sequence such that

and for each t

where “cl” denotes the closure in.

Proof. For each, by Lemma 7, there exists a sequence such that

where the closure is taken in.

For each, by Castaing represent theorem (cf.   ), there exists a sequence such that

At first we will show that

In fact, taking, there exists a sequence such that

then there exists a subsequence such that

Therefore

On the other hand

since is closed and, which yields

Since

is closed and, then for each t

(7)

For any, there exists a sequence such that

Then for each t,

which means

(8)

(7) together with (8) yields

Lemma 8. (cf.  ) Let, satisfy: for fixed is continuous with respect to x, for fixed, is measurable with respect to, then there exists an such that, then we have

Theorem 6. Let. Then for any,

Proof. Let. By Theorem 5, we can obtain that for each, there exist sequences and such that ,. For each t,

and

Therefore

(9)

By Lemma 8, we have

(10)

Then

(11)

Then

Similarly, we have

Then for each,

Therefore

Hence

Theorem 7. Let. Then for each we have

Proof. For any, we have

(12)

by Lemma 8, we have

(13)

Then

(14)

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.

4. Conclusion

In  , the author studied the Lebesgue-Stieltjes integral of real stochastic processes with respect to fuzzy valued stochastic processes. In some references such as   , 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, we defined directly the integral of fuzzy stochastic 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 -bounded under the metric (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.

Acknowledgements

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

Conflicts of Interest

The authors declare no conflicts of interest.

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