Almost sure exponential stability of the backward Euler–Maruyama scheme for stochastic delay differential equations with monotone-type condition

Abstract

This paper is a continuation of our previous paper, in which, the second author, with Mao and Szpruch examined the almost sure stability of the Euler–Maruyama (EM) and the backward Euler–Maruyama (BEM) methods for stochastic delay differential equations (SDDEs). In the previous results, although the drift coefficient may defy the linear growth condition, the diffusion coefficient is required to satisfy the linear growth condition. In this paper we want to further relax the condition. Under monotone-type condition, this paper will give the almost sure stability of the BEM for SDDEs whose both drift and diffusion coefficients may defy the linear condition. This improves the existing results considerably.