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 It\^o formula [Sun, Yang and Zhao, Numer. Math. Theor. Meth. Appl., 10 (2017), pp.~798--828], we develop the It\^o formula and construct the It\^o-Taylor expansion for MSDEJs. Then based on the It\^o-Taylor expansion, we propose the strong order $\gamma$ and the weak order $\eta$ It\^o-Taylor schemes for MSDEJs. %We theoretically prove The strong and weak convergence rates $\gamma$ and $\eta$ of the strong and weak It\^o-Taylor schemes are theoretically proved, respectively. Finally some numerical tests are also presented to verify our theoretical conclusions.