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). 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.