Variable Exponent Function Spaces related to a Sublinear Expectation

Abstract

In this paper, variable exponent function spaces , , and are introduced in the framework of sublinear expectation, and some basic and important properties of these spaces are given. A version of Kolmogorov’s criterion on variable exponent function spaces is proved for continuous modification of stochastic processes.

1. Introduction

Variable exponent spaces are extensively applied in the study of some nonlinear problems in natural science and engineering. Basic properties of the spaces are first given by Kováčik and Rákosník in [1]. Some theories of variable exponent spaces can also be found in [2, 3]. Harjulehto et al. present an overview of applications to differential equations with nonstandard growth in [4]. Diening et al. [5] summarize most of the existing literature of theory of variable exponent function spaces and applications to partial differential equations. In [6], Aoyama proves some important probability inequalities in variable exponent Lebesgue spaces.

Nonlinear expectations play an important role in the research of financial markets. One of the most important application is that a coherent risk measure(the basic theory about coherent risk measure can be found in [7]) is a sublinear expectation defined on , which is a linear space of finacial losses. In this paper, we are interested in behavior of sublinear expectation spaces with variable exponents. By the following representation theorem which can be found in Peng ([8], p. 4), we know that a sublinear expectation can be expressed as a supremum of linear expectations, i.e., there exists a family of linear expectations such that

Thus, we consider the upper expectation only in this paper. Some other important theories about nonlinear expectations can be found in Peng’s [9, 10].

The remainder of the paper is divided as follows: in Section 2, motivated by Fu [11] and Denis et al. [12], we first introduce , and and give some properties of these spaces. And Each element of -completion of has a quasi-continuous version is proved. In Section 3, applying the results of Section 2, we discuss a version of Kolmogorov’s criterion for continuous modification of stochastic processes, which are in the variable exponent function space related to a sublinear expectation, after proving the situation under a linear expectation.

2. Variable Exponent Function Spaces

Let be a complete metric space equipped with the distance , the Borel algebra of and the collection of all probability measures on . : the space of all -measurable real functions; : all bounded functions in ; : all continuous functions in .

For a given subset , we denote

Definition 1. A set is a polar if and a property holds “quasi-surely” (q.s.) if it holds outside a polar set.

The upper expectation of is defined as follows: for each such that exists for each ,

About , we know the following properties.

Theorem 2 (see Theorem 9 in [12]). The upper expectation of family is a sublinear expectation on as well as on , i.e.,

(1)For all , in, , .(2)For all , in, .(3)For all , , .(4)For all , , .

Let a -measurable real function , be a variable exponent. In the space , the moduli are defined by

Definition 3. The space is the set of satisfying , and it is endowed with the Luxemburg norm:

We set , and denote . As usual, we do not take care about the distinction between classes and their representatives.

Proposition 4. If the variable exponent satisfies , then the inequalityholds for every and , where .

Proof. By Young inequality, we have

By the monotonicity, sub-additivity and positive homogeneity of , and the property of the norm, we have

Thus, the inequality follows.

Proposition 5. Suppose that the variable exponent satisfies . If , then(1)If , then .(2)If , then .(3), if and only if (4), if and only if .In particular, the linear expectation also follows this proposition.

Proof. (1) By and the definition of the norm,

Thus, . As

we also have

That is to say The proof is completed. (2) can be easily proved in the similar method. By (1) and (2), we can easily get (3) and (4).

Proposition 6. Suppose that the variable exponent satisfies . Let . Then for each

Proof. For each , by Markov inequality, we have

Take the supremum in both sides,

Thus,

Lemma 7 (Proposition 17 in [12]). Let and be a sequence in which converges to in . Then there exists a subsequence which converges to quasi-surely in the sense that it converges to X outside a polar set.

Proposition 8. If the variable exponent satisfies . is a Banach space.

Proof. Let be a Cauchy sequence in . Then, by Proposition 4,where is a constant. That is to say is a Cauchy sequence in . By Proposition 14 in [12], is a Banach Space. Thus, converges in . Suppose that , and further by Lemma 7, we suppose quasi-surely (subtracting a subsequence if necessary). For each , there exists such that for . Fix , by Fatou’s lemmaThus, , and further , that is to say, , and . The proof is completed.

We denote by the completion of and by the completion of . By Proposition 8, we have for .

Proposition 9. Suppose that the variable exponent satisfies , then

Proof. We denote . For each let . We have

so . Thus, . On the other hand, for any , we can find a sequence such that . Let and . Since , we have . This implies that for any sequence such that as , . And for all ,

For the first term of right hand side, we have

For the second term,

so

and . Thus, .

Proposition 10. If the variable exponent satisfies , let , then for each , there exists a , such that for all with , we have .

Proof. For each , by Proposition 9, there exists such that . Set , then for with , we have.

Definition 11. A mapping on with values in topological space is said to be quasi-continuous (q.c.) if , there exists an open set with such that is continuous.

Definition 12. We say that has a quasi-continuous version if there exists a quasi-continuous with q.s.

Proposition 13. If the variable exponent satisfies . Then each element in has a quasi-continuous version.

Proof. For each , there exists such that in . Let us choose a subsequence such that

and set for all ,

Because of the subadditivity property and Proposition 6,

Thus, , so the Borel set is polar. As each is continuous, for all , is an open set. And for all , converges uniformly on so that the limit is continuous on each .

Proposition 14. Suppose that the variable exponent satisfies , then

Proof. Let

For each , has a quasi-continuous version by Proposition 13. Since , by Proposition 9 we have . Thus, .

On the other hand, Let be quasi-continuous. For all , define

Since and

we have .

For all , is quasi-continuous, so there exists a closed set such that and is continuous on . By Tietze’s extension theorem there exists such that and on . Then, we have

Thus,

As a consequence, .

Proposition 15. Suppose that has a quasi-continuous version and that there exists a function satisfying uniformly on and . Then , where .

Proof. For each , there exists which is independent of such that for all ,

So

Thus, .

3. Kolmogorov’s Criterion on Variable Exponent Function Spaces

Definition 16. Let be a set of indices. and be two processes indexed by . We say that is a quasi-modification of if for all q.s.

To prove a Kolmogorov’s criterion for a process indexed by with on variable exponent function spaces, we give the following lemma first.

Lemma 17. Let and be a process such that for all , . Assume that for a fixed there exist positive constants and such that

Then admit a modification such that

for every

Proof. Fix . For , define the set of dyadic points in :

and . Set , and there are fewer than such pairs in . For , we say that if each component of is less than or equal to the corresponding component of .

Set . And by the hypothesis in the lemma,

It is easy to see from some on.

For , there exists increasing sequences and such that , , and , from some on.

Let now and , and either or and in any case,

where the series are actually finite sums. It follows that

Thus, setting , we have

As

for every , we have

where . It follows in particular that for almost every , is uniformly continuous on and it makes sense to set

By Fatou’s lemma and the hypothesis,

so, a.s. and is a modification.

Theorem 18. Let and be a process such that for all , . Assume that there exists a positive constants and such thatThen admit a modification such thatfor every

Proof. Let set the same as the proof of Lemma 17. We set.

where . By Lemma 17, we know that for any , is finite and uniformly bounded with respect to so that

Thus, is uniformly continuous on q.s. and set

In the similar way in Lemma 17, q.s. and is a modification.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The author gratefully acknowledges the support from Fujian Provincial Natural Science Foundation (2016J05008).