On exponential mean-square stability of two-step Maruyama methods for stochastic delay differential equations

We are concerned with the exponential mean-square stability of two-step Maruyama methods for stochastic differential equations with time delay. We propose a family of schemes and prove that it can maintain the exponential mean-square stability of the linear stochastic delay differential equation for every step size of integral fraction of the delay in the equation. Numerical results for linear and nonlinear equations show that this family of two-step Maruyama methods exhibits better stability than previous two-step Maruyama methods.