Polynomial stability of exact solution and a numerical method for stochastic differential equations with time-dependent delay

TL;DR

This paper investigates the polynomial stability of exact solutions and a numerical method for stochastic differential equations with time-dependent delay, providing conditions for stability and illustrating results with examples.

Contribution

It introduces sufficient conditions for polynomial stability of both the exact solution and a modified Euler-Maruyama method with time-dependent delays.

Findings

Conditions for polynomial stability with bounded delays

Conditions for polynomial stability with unbounded delays

Numerical examples confirming theoretical results

Abstract

Polynomial stability of exact solution and modified truncated Euler-Maruyama method for stochastic differential equations with time-dependent delay are investigated in this paper. By using the well known discrete semimartingale convergence theorem, sufficient conditions are obtained for both bounded and unbounded delay $\delta$ to ensure the polynomial stability of the corresponding numerical approximation. Examples are presented to illustrate the conclusion.