Polynomial and exponential stability of $\theta$-EM approximations to a class of stochastic differential equations

TL;DR

This paper investigates the stability properties of $ heta$-Euler-Maruyama numerical solutions for stochastic differential equations, providing conditions for polynomial and exponential stability, and extending previous results to more general cases.

Contribution

It offers new sufficient conditions for stability of $ heta$-EM approximations, improving and generalizing earlier findings, with comprehensive examples and counterexamples.

Findings

Established conditions for polynomial stability

Derived criteria for exponential stability

Extended stability results to broader classes of equations

Abstract

Both the mean square polynomial stability and exponential stability of $\theta$ Euler-Maruyama approximation solutions of stochastic differential equations will be investigated for each $0\le\theta\le 1$ by using an auxiliary function $F$ (see the following definition (2.3)). Sufficient conditions are obtained to ensure the polynomial and exponential stability of the numerical approximations. The results in Liu et al [12] will be improved and generalized to more general cases. Several examples and non stability results are presented to support our conclusions.