Stability of analytical and numerical solutions of nonlinear stochastic delay differential equations

TL;DR

This paper establishes stability conditions for both analytical and numerical solutions of nonlinear stochastic delay differential equations, including variable delays, and shows that the backward Euler method preserves these stability properties without Lyapunov function construction.

Contribution

It provides a unified theoretical framework for stability analysis of SDDEs with constant and variable delays and demonstrates the stability-preserving property of the backward Euler method without Lyapunov function requirements.

Findings

Derived sufficient stability conditions for nonlinear SDDEs

Proved backward Euler method preserves stability properties

Results applicable to SDDEs with bounded and unbounded delays

Abstract

This paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations (SDDEs). We derive sufficient conditions for the stability, contractivity and asymptotic contractivity in mean square of the solutions for nonlinear SDDEs. The results provide a unified theoretical treatment for SDDEs with constant delay and variable delay (including bounded and unbounded variable delays). Then the stability, contractivity and asymptotic contractivity in mean square are investigated for the backward Euler method. It is shown that the backward Euler method preserves the properties of the underlying SDDEs. The main results obtained in this work are different from those of Razumikhin-type theorems. Indeed, our results hold without the necessity of constructing of finding an appropriate Lyapunov functional.