On Halanay-type analysis of exponential stability for the θ-Maruyama method for stochastic delay differential equations

Using an approach that has its origins in work of Halanay, we consider stability in mean square of numerical solutions obtained from the θ-Maruyama discretization of a test stochastic delay differential equation dX(t)={f(t)-aX(t)+bX(t-\tau)}dt + {g(t)+\eta X(t)+\mu X(t-\tau)}dW(t), interpreted in the Itô sense, where W(t) denotes a Wiener process. We focus on demonstrating that we may use techniques advanced in a recent report by Baker and Buckwar to obtain criteria for asymptotic and exponential stability, in mean square, for the solutions of the recurrence X_{n+1}-X_n = θ h {f_{n+1} -a X_{n+1} +b X_{n+1-N}} + (1- θ h) {f_{n} -a X_{n} +b X_{n-N}} + \qrt{h} {g_n + \eta X_n + \mu X_{n-N} \xi_n, \xi_n \in N(0,1).