Numerical methods for mean-field stochastic differential equations with jumps

Abstract

In this paper, we are devoted to the numerical methods for mean-field stochastic differential equations with jumps (MSDEJs). By combining with the mean-field Itô formula (see Sun, Yang, and Zhao, Numer. Math. Theor. Meth. Appl., 10, pp. 798–828 (2017)), we first develop the Itô formula and further construct the Itô-Taylor expansion for MSDEJs. Then based on the Itô-Taylor expansions, we propose the strong order γ and the weak order η Itô-Taylor schemes for MSDEJs. We theoretically prove the strong convergence rate γ of the strong order γ Itô-Taylor scheme and the weak convergence rate η of the weak order η Itô-Taylor scheme, respectively. Some numerical tests are also presented to verify our theoretical conclusions.

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Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions which helped to improve much of the quality of the paper.

Funding

This research is partially supported by the NSF of China (Nos. 12071261, 12001539, 11831010, 11871068), the Science Challenge Project (No. TZ2018001), the national key basic research program (No. 2018YFA0703903, No. 2018YFB0704304), the NSF of Hunan Province (No. 2020JJ5647), and China Postdoctoral Science Foundation (No. 2019TQ0073).

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Correspondence to Weidong Zhao.

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Sun, Y., Zhao, W. Numerical methods for mean-field stochastic differential equations with jumps. Numer Algor (2021). https://doi.org/10.1007/s11075-020-01062-w

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Keywords

  • Mean-field stochastic differential equations with jumps
  • Itô formula
  • Itô-Taylor expansion
  • Itô-Taylor schemes
  • Error estimates

Mathematics Subject Classification (2010)

  • 60H35
  • 65C20
  • 60H10