Convergence rate of Euler\u2013Maruyama scheme for SDDEs of neutral type

Yanting Ji

doi:10.1186/s13660-020-02533-3

Abstract

In this paper, we are concerned with the convergence rate of Euler–Maruyama (EM) scheme for stochastic differential delay equations (SDDEs) of neutral type, where the neutral, drift, and diffusion terms are allowed to be of polynomial growth. More precisely, for SDDEs of neutral type driven by Brownian motions, we reveal that the convergence rate of the corresponding EM scheme is one-half; Whereas for SDDEs of neutral type driven by pure jump processes, we show that the best convergence rate of the associated EM scheme is slower than one-half. As a result, the convergence rate of general SDDEs of neutral type, which is dominated by pure jump process, is slower than one-half.

Highlights

There is numerous literature concerned with convergence rate of numerical schemes for stochastic differential equations (SDEs)
Under the Khasminskii-type condition, Mao [11] revealed that the convergence rate of the truncated EM method is close to one-half; under the Hölder condition, the convergence rate of EM scheme for SDEs has been studied by many scholars; Sabanis [19] recovered the classical rate of convergence for the tamed EM schemes, where, for the SDE involved, the drift coefficient satisfies a onesided Lipschitz condition and a polynomial Lipschitz condition, and the diffusion term is Lipschitzian
Under a log-Lipschitz condition, Bao et al [5] studied the convergence rate of EM approximation for a range of stochastic functional differential equations (SFDEs) driven by jump processes; Bao and Yuan [4] investigated the convergence rate of EM approach for a class of stochastic differential delay equations (SDDEs), where the drift and diffusion coefficients are allowed to be of

Summary

Introduction

There is numerous literature concerned with convergence rate of numerical schemes for stochastic differential equations (SDEs). To the best of our knowledge, the convergence rate of explicit EM scheme for SFDEs of neutral type with non-Lipschitz conditions ( nonlinear) has seen few results.

Objectives

Results

Conclusion