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The strong convergence and stability of explicit approximations for nonlinear stochastic delay differential equations

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Abstract

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under less restrictive conditions, the truncated Euler-Maruyama (TEM) schemes for SDDEs are proposed, which numerical solutions are bounded in the q th moment for q ≥ 2 and converge to the exact solutions strongly in any finite interval. The 1/2 order convergence rate is yielded. Furthermore, the long-time asymptotic behaviors of numerical solutions, such as stability in mean square and \(\mathbb {P}-1\), are examined. Several numerical experiments are carried out to illustrate our results.

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Acknowledgements

The authors thank the associated editor and referee for the helpful comments and suggestions.

Funding

The research of Junhao Hu was supported by the National Natural Science Foundation of China (No. 61876192). The research of Xiaoyue Li was supported by the National Natural Science Foundation of China (No. 11971096), the Natural Science Foundation of Jilin Province (No. YDZJ202101ZYTS154), the Education Department of Jilin Province (No. JJKH20211272KJ), and the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Guoting Song & Xiaoyue Li

  2. School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China

    Junhao Hu & Shuaibin Gao

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  1. Guoting Song
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  2. Junhao Hu
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  3. Shuaibin Gao
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  4. Xiaoyue Li
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Correspondence to Xiaoyue Li.

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Song, G., Hu, J., Gao, S. et al. The strong convergence and stability of explicit approximations for nonlinear stochastic delay differential equations. Numer Algor 89, 855–883 (2022). https://doi.org/10.1007/s11075-021-01137-2

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  • DOI: https://doi.org/10.1007/s11075-021-01137-2

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