The th moment stability for the stochastic pantograph equation

In this paper, we investigate the α th moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the Razumikhin technique. Especially the linear stochastic pantograph equations and the semi-implicit Euler method applying them are considered. The convergence result of the semi-implicit Euler method is obtained. The stability conditions of the analytic solution of those equations and the numerical method are given. Finally, some experiments are given.