Convergence and stability of a numerical method for nonlinear stochastic pantograph equations

Stochastic pantograph equations (SPEs) are a special case of stochastic delay differential equations (SDDEs) with unbounded memory. Here we examine analytical and numerical solutions of nonlinear SPEs. Some sufficient conditions for the mean-square (MS) stability of the analytical solutions are obtained. We consider the Euler-Maruyama (EM) method and develop a fundamental analysis concerning its strong convergence and MS stability. We show that the numerical solution produced by the EM method converges to the exact solution under the local Lipschitz condition. We find that the stability conditions for the EM method are somewhat stronger than those for the analytical solution. Therefore, we consider an implicit method, the backward Euler (BE) method. By constructing an ^Ft-adaptive continuous approximate solution, we obtain an analogous result of convergence under the local Lipschitz condition, that is, the BE solution converges to the exact solution. Furthermore, we prove that if the nonlinear SPEs are stable, then so is the BE method applied to the systems for any step size. Some numerical examples are presented to demonstrate the MS stability of the numerical methods.