Symmetric Fuzzy Stochastic Differential Equations with Generalized Global Lipschitz Condition

Marek T Malinowski

doi:10.3390/sym12050819

Abstract

The paper contains a discussion on solutions to symmetric type of fuzzy stochastic differential equations. The symmetric equations under study have drift and diffusion terms symmetrically on both sides of equations. We claim that such symmetric equations have unique solutions in the case that equations’ coefficients satisfy a certain generalized Lipschitz condition. To show this, we prove that an approximation sequence converges to the solution. Then, a study on stability of solution is given. Some inferences for symmetric set-valued stochastic differential equations end the paper.

Highlights

Stochastic differential equation are natural mathematical tools to describe behavior of many dynamic systems evolving in time
In [17,18,19], we considered such equations in their natural integral form, which is a direct reflection of the form of crisp stochastic differential equations, i.e., x ( t ) = x0 ⊕
= y ( t ), Before we begin a deeper theoretical analysis of the task posed in this paper, we present an example illustrating the motivation to study fuzzy stochastic differential equations in their symmetric form

Summary

Introduction

Stochastic differential equation are natural mathematical tools to describe behavior of many dynamic systems evolving in time. Symmetry 2020, 12, 819 where f is a fuzzy stochastic mapping, g is a single-valued stochastic mapping, and x0 is a fuzzy random variable The form of this equation is asymmetric because more components are on the right side of the equation. They are the first fundamental step towards possibility of future research on periodic solutions of fuzzy stochastic differential equations. Neither of the first two equations can have periodic solutions due to the property of monotonicity of fuzziness in successive values These new symmetric equations do not contain this inconvenience. With the help of this sequence, existence of the unique solution to symmetric fuzzy stochastic differential equations is proved. A conclusion in Section 6 summarizes the contribution of the paper

Preliminaries

Unique Solutions

Stability of Solution

Application to Symmetric Multivalued Stochastic Differential Equations

Conclusions