Multi-step Maruyama methods for stochastic delay differential equations

Abstract

In this paper the numerical approximation of solutions of It{\^o} stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their $L_p$-consistency, numerical $L_p$-stability and $L_p$-convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.