Asymptotic Stability Of Explicit Runge - Kutta Methods For Stochastic Delay Differential Equations

Most of stochastic delay differential equation can not be solved analytically so that numerical methods is one of the important ways to investigate the dynamics of its solution. As we know,Runge-Kutta method is usually applied to solving stochastic differential equation. However, until now, relative little is known about Runge-Kutta method for solving stochastic delay differential equation. On the other hand, the stability of an explicit numerical scheme is not as good as the implicit one, but the explicit one has better computational e?ciency. Further, a kind of explicit Runge-Kutta method based on Chebyshev polynomial shows good stability properties, and its stability domain can be extended by increasing the stage number of the Runge-Kutta method.This paper focuses on the stability analysis of the explicit Runge-Kutta Chebyshev method applied to stochastic delay differential equation. We analyzed the mean square stability of scalar and system S-ROCK method. By comparing with the explicit Euler-Maruyama scheme, the explicit Runge-Kutta Chebyshev method shows better stability property. Numerical examples confirm our theoretical analysis.