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.

在本文中，我们致力于具有跳跃的平均场随机微分方程（MSDEJ）的数值方法。通过结合平均场Itô公式（请参见Sun，Yang和Zhao，Numer。Math。Theor。Meth。Appl。，10，pp。798-828（2017）），我们首先开发Itô公式并进一步构建MSDEJ的Itô-Taylor扩展。然后根据伊藤泰勒展开，我们提出了强烈的秩序γ和弱势整理η伊藤泰勒方案为MSDEJs。从理论上证明了强收敛速度γ阶强者γ伊藤泰勒方案和弱收敛速度η弱势整理的ηItô-Taylor方案。还提出了一些数值测试，以验证我们的理论结论。