A Note on Exponential Stability for Numerical Solution of Neutral Stochastic Functional Differential Equations

Qi Wang,Huabin Chen,Chenggui Yuan

doi:10.3390/math10060866

Qi Wang, Huabin Chen + Show 1 more

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https://doi.org/10.3390/math10060866

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Journal: Mathematics	Publication Date: Mar 9, 2022
Citations: 2	License type: CC BY 4.0

Affiliation: Nanchang University, Swansea University

Abstract

This paper examines the numerical solutions of the neutral stochastic functional differential equation. This study establishes the discrete stochastic Razumikhin-type theorem to investigate the exponential stability in the mean square sense of the Euler–Maruyama numerical solution to this equation. In addition, the Borel–Cantelli lemma and the stochastic analysis theory are incorporated to discuss the almost sure exponential stability for this numerical solution of such equations.

Highlights

Some realistic dynamical systems are concerned with the present and past states and the derivatives of the past states, which are mathematically characterized by neutral functional differential equations (NFDEs) in [1]
When environmental perturbation is considered, NFDEs are developed into neutral stochastic functional differential equations (NSFDEs); for more details, see [2] and the references therein
The existence, uniqueness, stability analysis, and boundedness for the solutions of NSFDEs have been investigated over the past few decades; see [3–6] and references therein

Summary

Introduction

Some realistic dynamical systems are concerned with the present and past states and the derivatives of the past states, which are mathematically characterized by neutral functional differential equations (NFDEs) in [1]. In [6], by establishing the stochastic version of the Razumikhin-type theorem, the exponential stability in moment for NSFDEs was investigated, and under one additional condition, the almost surely exponential stability was obtained. In [11], by establishing the discrete Razumikhintype theorem, the exponential stability in moment and the almost sure exponential stability of the EM scheme of SFDEs were investigated. In [15], the discrete stochastic version of the Razumikhin-type theorem was used to analyze the exponential stability in moment, the almost sure exponential stability for the EM scheme, and the backward EM scheme of nonlinear stochastic pantograph differential equations. As far as we know, there is no work on the discrete stochastic version of Razumikhin-type theorem to analyze the stability of the EM scheme for NSFDEs. In this paper, we try to close this gap.

Notations and Preliminaries

The EM Numerical Solutions of NSFDEs

Stability of Numerical Solutions of NSFDEs