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

Christopher T.H. Baker , Evelyn Buckwar

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 14

MSC 2000

65C30 Stochastic differential and integral equations

60H35 Computational methods for stochastic 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 {\em $\theta$--Maruyama discretization} of a test stochastic delay differential equation $$ dX(t) = \{f(t) -\alpha X(t) + \beta X(t-\tau)\} {\rd}t + \{g(t) + \eta \ X(t) + \mu X (t-\tau) \} \ {\rd}W(t), $$ interpreted in the It\^o 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 $$ {\wtX}_{n+1} \!-\! {\wtX}_n \!=\! \theta h\{f_{n+1} \!-\! \A {\wtX}_{n+1} \!+\! \B {\wtX}_{n+1-N}\} \ + $$ $$ \! + (1-\theta) h \{f_n \!-\! \A {\wtX}_{n} \! +\! \B {\wtX}_{n-N}\} \!+\! \sqrt{h}(g_n \!+\! \eta {\wtX}_n \!+\! \mu {\wtX}_{n-N_{}})\xi_n\quad (\xi_n \in \N (0,1)). $$