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

The asymptotic mean-square stability of two-step Maruyama methods is considered for nonlinear neutral stochastic differential equations with constant time delay (NSDDEs). Under the one-sided Lipschitz condition and the linear growth condition, it is proved that a family of implicit two-step Maruyama methods can preserve the stability of the analytic solution in mean-square sense. Numerical results for both a nonlinear NSDDE and a system show that the family of two-step Maruyama methods have better stability than previous two-step Maruyama methods.