This paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.

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<p style='text-indent:20px;'>In this work, by combining with stochastic approximation methods, we proposed a new explicit multistep scheme for solving the forward backward stochastic differential equations. Compared with the one constructed by using derivative approximation method, the new one covers the approximation of the stochastic part and is more accurate and easier to realize. Several numerical tests are presented to show the stability and effectiveness of the proposed scheme.</p>

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