Strong convergence of the split-step backward Euler method for stochastic delay differential equations with a nonlinear diffusion coefficient

Abstract

This paper is concerned with the strong convergence of numerical methods for stochastic differential equations with a variable delay. Based on the concepts of stochastic C-stability and B-consistency, we first derive a fundamental strong convergence theorem for general one-step methods applied to nonlinear autonomous stochastic delay differential equations under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. These conditions admit that the diffusion coefficient is highly nonlinear. Then, we construct the split-step backward Euler method. It is verified that the method is stochastically C-stable and B-consistent of order 1/2 so that the method is strongly convergent of order 1/2 for the considered problems. Finally, the theoretical results are confirmed by numerical experiments.