Advances in the truncated Euler–Maruyama method for stochastic differential delay equations

Weiyin Fei; Liangjian Hu; Xuerong Mao; Dengfeng Xia

doi:10.3934/cpaa.2020092

Article Contents

2020, Volume 19, Issue 4: 2081-2100. Doi: 10.3934/cpaa.2020092

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Advances in the truncated Euler–Maruyama method for stochastic differential delay equations

1.
Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, Anhui 241000, China
2.
Department of Applied Mathematics, Donghua Univerisity, Shanghai 201620, China
3.
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, U.K
4.
School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, Anhui 241000, China

Dedicated to Prof. Dr. Tomás Caraballo's 60th birthday

Received: January 31, 2019

Revised: June 30, 2019

Published: January 2020

This research was partially supported by the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal SocietyNewton Advanced Fellowship), the EPSRC (EP/K503174/1), the National Natural Science Foundation of China (61673103, 71571001), the Natural Science Foundation of Shanghai (17ZR1401300) and the Ministry of Education (MOE) of China (MS2014DHDX020)

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Abstract

Guo et al. [8] are the first to study the strong convergence of the explicit numerical method for the highly nonlinear stochastic differential delay equations (SDDEs) under the generalised Khasminskii-type condition. The method used there is the truncated Euler–Maruyama (EM) method. In this paper we will point out that a main condition imposed in [8] is somehow restrictive in the sense that the condition could force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is then to establish the convergence rate without this restriction.

Keywords:

Brownian motion, stochastic differential delay equation,
Itô's formula,
truncated Euler–Maruyama,
Khasminskii-type condition.

Mathematics Subject Classification: Primary: 60H10, 60J65.

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References

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