Itô-Taylor Schemes for Solving Mean-Field Stochastic Differential Equations

Abstract

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 meanfield SDE setting. Finally some numerical examples are given to verify our theoretical results.