Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations

In this paper, we establish Razumikhin-type theorems on th moment polynomial stability of exact solution for the stochastic pantograph differential equations, which improves the existing stochastic Razumikhin-type theorems. By using discrete Razumikhin-type technique, we construct conditions for the stability of general numerical scheme of the stochastic pantograph differential equations (SPDEs). The stabilities mainly conclude the global th moment asymptotically stability and th moment polynomial stability. Using the conditions constructed for the stability of the numerical solutions, we discuss the stability of two special numerical methods, namely the Euler-Maruyama method and the backward Euler-Maruyama method. Finally, an example is given to illustrate the consistence with the theoretical results on th moment polynomial stability.