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). 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. 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.