On stability of numerical solutions of neutral stochastic delay differential equations with time‐dependent delay

Abstract

This paper focuses on exponential stability of numerical solutions of neutral stochastic delay differential equations. A novel approach is introduced to study numerical approximation methods of neutral stochastic differential equations with time‐dependent delay. In contrast to the advances in the literature, this work provides a new method and new criteria for which the Euler‐Maruyama approximation method and the backward Euler‐Maruyama approximation method can reproduce exponential stability in mean square and almost sure exponential stability for sufficiently small step sizes. Two examples are provided to demonstrate the effectiveness of our criteria.