Convergence Rate of EM Scheme for SDDEs

TL;DR

This paper analyzes the convergence rate of the Euler-Maruyama scheme for stochastic differential delay equations with nonlinear coefficients, establishing a rate of 1/2 for Brownian motion and near 1/2 for jump processes.

Contribution

It provides the first detailed analysis of convergence rates for Euler-Maruyama applied to SDDEs with highly nonlinear delay coefficients.

Findings

Convergence rate is 1/2 for SDDEs driven by Brownian motion.

Mean-square convergence is optimal for jump processes.

Order of mean-square convergence is close to 1/2.

Abstract

In this paper we investigate the convergence rate of Euler-Maruyama scheme for a class of stochastic differential delay equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the convergence rate of Euler-Maruyama scheme is 1/2$ for the Brownian motion case, while show that it is best to use the mean-square convergence for the pure jump case, and that the order of mean-square convergence is close to 1/2.