Mean square convergence of stochastic theta-methods for nonlinear neutral stochastic differential delay equations
S. Gan 1, H. Schurz 2, H. Zhang 11 School of Mathematical Sciences and Computing Technology, Central South University, Changsha 410075, Hunan, China.
2 Department of Mathematics, Southern Illinois University, Mailcode 4408, 1245 Lincoln Drive, Carbondale, IL 62901, USA.
Received by the editors June 14, 2010 and ,in revised form, October 29, 2010.
This paper is devoted to the convergence analysis of stochastic theta-methods for nonlinear neutral stochastic differential delay equations (NSDDEs) in Ito 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.AMS subject classifications: 65C30, 60H10, 60H35, 65C20
Key words: neutral stochastic differential delay equations, mean-square continuity, stochastic theta-methods, mean-square convergence, consistency.