The split-step backward Euler method for linear stochastic delay differential equations

In this paper, the numerical approximation of solutions of linear stochastic delay differential equations (SDDEs) in the Itô sense is considered. We construct split-step backward Euler (SSBE) method for solving linear SDDEs and develop the fundamental numerical analysis concerning its strong convergence and mean-square stability. It is proved that the SSBE method is convergent with strong order γ=12 in the mean-square sense. The conditions under which the SSBE method is mean-square stable (MS-stable) and general mean-square stable (GMS-stable) are obtained. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square stability of the SSBE method.