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Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments

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Abstract

This work develops the Milstein scheme for commutative stochastic differential equations with piecewise continuous arguments (SDEPCAs), which can be viewed as stochastic differential equations with time-dependent and piecewise continuous delay. As far as we know, although there have been several papers investigating the convergence and stability for different numerical methods on SDEPCAs, all of these methods are Euler-type methods and the convergence orders do not exceed 1/2. Accordingly, we first construct the Milstein scheme for SDEPCAs in this work and then show its convergence order can reach 1. Moreover, we prove that the Milstein method can preserve the stability of SDEPCAs. In the last section, we provide several numerical examples to verify the theoretical results.

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Acknowledgements

The authors would like to thank the Journal Editorial Office Assistant, Jude Estrera, for helping in the preparation of this manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (Grant nos.12071101 and 11671113).

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Authors and Affiliations

  1. School of Mathematics, Harbin Institute of Technology, Xidazhi Street, Harbin, 150001, Heilongjiang, China

    Yuhang Zhang, Minghui Song & Mingzhu Liu

  2. Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Xidazhi Street, Harbin, 150001, Heilongjiang, China

    Bowen Zhao

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  1. Yuhang Zhang
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  2. Minghui Song
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  3. Mingzhu Liu
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  4. Bowen Zhao
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Yuhang Zhang drafted the manuscript, and all the authors revised the manuscript together.

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Correspondence to Minghui Song.

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Zhang, Y., Song, M., Liu, M. et al. Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments. Numer Algor 96, 417–448 (2024). https://doi.org/10.1007/s11075-023-01652-4

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