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

Abstract

Using an approach that has its origins in work of Halanay, we consider stability in mean square of numerical solutions obtained from the theta-Maruyama discretization of a test stochastic delay differential equation dX(t) = {f(t) -alpha X(t) + beta X(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 Xn+1 - Xn = theta h {fn+1 - alpha Xn+1 + beta Xn+1 - N} + + (1-theta) h {fn - alpha Xn + beta Xn-N} + sqrt{h} (gn + eta Xn + mu Xn-N) xi n (xi n in N (0,1)).