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Numerical Algorithms

, Volume 78, Issue 2, pp 599–624 | Cite as

The truncated Euler–Maruyama method for stochastic differential delay equations

  • Qian Guo
  • Xuerong Mao
  • Rongxian Yue
Open Access
Original Paper

Abstract

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512–5524 2011), and the theory there showed that the Euler–Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L p ) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370–384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.

Keywords

Brownian motion Stochastic differential delay equation Itô’s formula Truncated Euler–Maruyama Khasminskii-type condition 

Mathematical subject classifications (2000)

60H10 60J65 

Notes

Acknowledgements

The authors would like to thank the associate editor and referees for their very helpful comments and suggestions. The authors would also like to thank the Leverhulme Trust (RF-2015-385), the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the Natural Science Foundation of China (11471216), the Natural Science Foundation of Shanghai (14ZR1431300), the Ministry of Education (MOE) of China (MS2014DHDX020) for their financial support.

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Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Department of Mathematics and StatisticsUniversity of StrathclydeGlasgowUK

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