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Mean-square convergence and stability of two-step Milstein methods for stochastic differential equations with Poisson jumps

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Abstract

This paper deals with a class of two-step Milstein methods for stochastic differential equations with Poisson jumps. The mean-square convergence and linear mean-square stability of the proposed methods are discussed. In addition, the linear mean-square stability regions of the two-step Milstein methods are compared with those of one-step \(\theta \)-Milstein methods. Numerical examples demonstrate the mean-square convergence and the linear mean-square stability of the presented methods.

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Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the paper.

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Authors and Affiliations

  1. College of Science, Henan University of Technology, Zhengzhou, 450001, Henan, People’s Republic of China

    Quanwei Ren

  2. Department of Mathematics, Shanghai Normal University, 100 Guilin Road, Shanghai, 200234, People’s Republic of China

    Hongjiong Tian

  3. Scientific Computing Key Laboratory of Shanghai Universities, 100 Guilin Road, Shanghai, 200234, People’s Republic of China

    Hongjiong Tian

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  1. Quanwei Ren
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  2. Hongjiong Tian
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Correspondence to Quanwei Ren.

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Communicated by Pierre Etore.

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The work of Q. Ren is supported by National Natural Science Foundation of China (no. 11801146), The youth backbone teacher cultivation project of Henan University of Technology (21420123), The youth support project for basic research of Henan University of Technology (2018QNJH17) and the High-Level Personal Foundation of Henan University of Technology (no. 2017BS023). The work of H. Tian is supported in part by the National Natural Science Foundation of China under Grant nos. 11671266 and 11871343, and Science and Technology Innovation Plan of Shanghai under Grant no. 20JC1414200.

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Ren, Q., Tian, H. Mean-square convergence and stability of two-step Milstein methods for stochastic differential equations with Poisson jumps. Comp. Appl. Math. 41, 125 (2022). https://doi.org/10.1007/s40314-022-01824-3

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  • DOI: https://doi.org/10.1007/s40314-022-01824-3

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