On stability of solutions of stochastic delay differential equations

Abstract

This paper addresses exponential stability of a class of stochastic delay differential equations and their numerical solutions. We begin by establishing criteria for exponential stability in mean square of stochastic delay differential equations. We then show that the Euler–Maruyama approximation method correctly reproduces exponential stability in mean square for sufficiently small step sizes. Consequently criteria for almost sure exponential stability of the exact and the Euler–Maruyama approximation are also obtained. In contrast to the advances in the literature, this work provides more accurate estimates of the Lyapunov exponents of stochastic delay differential equations and their Euler–Maruyama approximations.