Numerical methods for mean-field stochastic differential equations with jumps

In this paper, we are devoted to the numerical methods for mean-field stochastic differential equations with jumps (MSDEJs). First by using the mean-field Itô formula [Sun, Yang and Zhao, Numer. Math. Theor. Meth. Appl., 10 (2017), pp.~798--828], we develop the Itô formula and construct the Itô-Taylor expansion for MSDEJs. Then based on the Itô-Taylor expansion, we propose the strong order $γ$ and the weak order $η$ Itô-Taylor schemes for MSDEJs. %We theoretically prove The strong and weak convergence rates $γ$ and $η$ of the strong and weak Itô-Taylor schemes are theoretically proved, respectively. Finally some numerical tests are also presented to verify our theoretical conclusions.

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