Asymptotic mean-square stability of explicit Runge–Kutta Maruyama methods for stochastic delay differential equations

As relatively little is known about Runge–Kutta type method applied to stochastic delay differential equations, we present an explicit Runge–Kutta Maruyama (RKM) method for solving them. The mean-square stability properties of the numerical solutions generated by the RKM method are investigated, and a sufficient condition for stability is obtained and applied to the S-ROCK type methods for stochastic delay differential equations. Numerical examples are provided to confirm theoretical results.