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

Jinping Zhang,Xingmei Li,Xiaoying Wang,Lingli Luo

doi:10.4236/am.2015.613193

Jinping Zhang, Xingmei Li + Show 2 more

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https://doi.org/10.4236/am.2015.613193

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Journal: Applied mathematics	Publication Date: Jan 1, 2015
Citations: 11	License type: CC BY 4.0

Affiliation: North China Electric Power University

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.

Highlights

The theory of fuzzy functions has been developed quickly due to the measurements of various uncertainties arising from the randomness and from the vagueness in some situations
As usual, in order to explore the integrals of fuzzy stochastic processes, at first we can study the integrals of set-valued stochastic processes
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

Summary

Introduction

The theory of fuzzy functions has been developed quickly due to the measurements of various uncertainties arising from the randomness and from the vagueness in some situations. 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 L2 -boundedness which are necessary to our future work on fuzzy stochastic differential equations

Preliminaries

Lebesgue-Stieltjes Integrals with Respect to Finite Variation Processes

Conclusion