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)− αX(t) + βX(t− τ )}dt+ {g(t) + η X(t) + μX(t− τ )} 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 e Xn+1− e Xn =θh{fn+1−α e Xn+1+β e Xn+1−N} + +(1− θ)h{fn−α e Xn+β e Xn−N}+ √ h(gn+η e Xn+μ e Xn−N )ξn (ξn ∈ N (0, 1)). θ-Maruyama scheme; asymptotic and exponential stability; stochastic delay differential & difference equations; Halanay-type inequalities. AMS Subject Classification: 65C30 60H35 34K20 34K50