Convergence rates of theta-method for NSDDEs under non-globally Lipschitz continuous coefficients

Li Tan,Chenggui Yuan

doi:10.1142/s1664360719500061

Abstract

This paper is concerned with strong convergence and almost sure convergence for neutral stochastic differential delay equations under non-globally Lipschitz continuous coefficients. Convergence rates of [Formula: see text]-EM schemes are given for these equations driven by Brownian motion and pure jumps, respectively, where the drift terms satisfy locally one-sided Lipschitz conditions, and diffusion coefficients obey locally Lipschitz conditions, and the corresponding coefficients are highly nonlinear with respect to the delay terms.

Highlights

With the development of computer technology, numerical analyses have witnessed rapid growth since most equations cannot be solved explicitly
As to its numerical analysis, Wu and Mao [16] examined numerical solutions of neutral stochastic functional differential equations and established the strong mean square convergence theory of EM scheme under local Lipschitz condition; Zhou [17] established a criterion on exponential stability of EM scheme and backward scheme to NSDDES; Zong and Huang [19] were concerned with pth moment and almost sure exponential stability of the exact and EM-scheme solutions of Neutral stochastic differential delay equations (NSDDEs); Ji et al [7] generalized the results of [1] to NSDDEs; Tan and Yuan [15] proposed a tamed θ-EM scheme and gave convergence rate for NSDDEs driven by Brownian motion and pure jumps under onesided Lipschitz condition
The rest of paper is organized as follows: in Sec. 2, strong convergence rate and almost sure convergence rate are given for NSDDEs driven by Brownian motion under nonglobally Lipschitz condition, while in Sec. 3, the Brownian motion is replaced by pure jumps, under similar conditions, the convergence rates are provided

Summary

Introduction

With the development of computer technology, numerical analyses have witnessed rapid growth since most equations cannot be solved explicitly. As to its numerical analysis, Wu and Mao [16] examined numerical solutions of neutral stochastic functional differential equations and established the strong mean square convergence theory of EM scheme under local Lipschitz condition; Zhou [17] established a criterion on exponential stability of EM scheme and backward scheme to NSDDES; Zong and Huang [19] were concerned with pth moment and almost sure exponential stability of the exact and EM-scheme solutions of NSDDEs; Ji et al [7] generalized the results of [1] to NSDDEs; Tan and Yuan [15] proposed a tamed θ-EM scheme and gave convergence rate for NSDDEs driven by Brownian motion and pure jumps under onesided Lipschitz condition. The rest of paper is organized as follows: in Sec. 2, strong convergence rate and almost sure convergence rate are given for NSDDEs driven by Brownian motion under nonglobally Lipschitz condition, while in Sec. 3, the Brownian motion is replaced by pure jumps, under similar conditions, the convergence rates are provided

Preliminaries

Moment bounds

Convergence rates

Convergence Rates for Pure Jumps Case