Mean Square Convergence of Stochastic $\theta$-Methods for Nonlinear Neutral Stochastic Differential Delay Equations

Abstract

This paper is devoted to the convergence analysis of stochastic $\theta$-methods for nonlinear neutral stochastic differential delay equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of the solution in the past values, which leads to a stochastic differential algebraic system. Drift-implicit stochastic $\theta$-methods are proposed for the coupled system. It is shown that the stochastic $\theta$-methods are mean-square convergent with order 1/2 for Lipschitz continuous coefficients of underlying NSDDEs. A nonlinear numerical example illustrates the theoretical results.