On strong convergence of explicit numerical methods for stochastic delay differential equations under non-global Lipschitz conditions

Abstract

In this paper, we study the convergence of explicit numerical methods in strong sense for stochastic delay differential equations (SDDEs) with super-linear growth coefficients. Under non-globally Lipschitz conditions, a fundamental theorem on convergence has been constructed to elaborate the relationship of convergence rate between the local truncated error and the global error of one-step explicit methods in the sense of pth moments. A class of balanced Euler schemes has been presented and the boundedness of numerical solutions has been proved. By using the fundamental theorem, we prove that the balanced Euler scheme is of 0.5 order convergence in mean-square sense. Numerical examples verify the theoretical predictions.