Split-step θ-method for stochastic delay differential equations

In this paper we study the mean-square stability and convergence of the split-step θ-method for stochastic differential equations with fixed time delay. Under mild assumptions, the split-step θ-method is proved to be exponentially mean-square stable and converge with strong order 1/2. Numerical examples show how mean-square stability of the split-step θ-method depends on the parameter θ and the step size h for both linear and nonlinear models.