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

In this paper, we establish two different alpha th moment psi-type stability criterions on the exact and numerical solutions of the stochastic pantograph differential equations. One is established by the virtue of Lyapunov method, and the other is used by the continuous and discrete Razumikhin technique. By comparing the two stability criterions, we can easily see that the conditions constructed by the Razumikhin-type technique are better than those constructed by the virtue of Lyapunov method. Using the conditions constructed for the ath moment psi-type stability, we study the stability of the Euler-Maruyama method and the backward Euler-Maruyama method, respectively, for a special class of the stochastic pantograph differential equations. Finally, examples are given to illustrate the consistence with the theoretical results on the ath moment psi-type stability. (C) 2019 Elsevier B.V. All rights reserved.