Stability of numerical solutions for the stochastic pantograph differential equations with variable step size

Abstract

In this paper, two different techniques are used to research the stability of numerical solutions with variable step size for stochastic pantograph differential equations. Based on the discrete semimartingale convergence theorem, the almost sure polynomial stability and the αth moment polynomial stability of the numerical solutions by the Euler–Maruyama (EM) and the backward Euler–Maruyama (BEM) methods with variable step size have been discussed. By using the discrete Razumikhin-type technique, the global αth moment asymptotically stability and the αth moment polynomial stability of the numerical solutions by the general numerical scheme with variable step size have been researched. Based on the results of the stability of the general numerical scheme, we discuss the stability of two special numerical solutions, namely the EM method and the BEM method. By comparing the two techniques (i.e. the discrete semimartingale convergence theorem and the discrete Razumikhin-type technique) on the research of the moment stability, we can easily get the latter is better. To illustrate the theoretical results, we give some examples to examine the almost sure polynomial stability and the αth moment polynomial stability of the numerical solutions by the EM and BEM schemes with variable step size.