TY - JOUR T1 - Itô-Taylor Schemes for Solving Mean-Field Stochastic Differential Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 798 EP - 828 PY - 2017 DA - 2017/10 SN - 10 DO - http://doi.org/10.4208/nmtma.2017.0007 UR - https://global-sci.org/intro/article_detail/nmtma/10457.html KW - Itô-Taylor scheme, mean-field stochastic differential equation, mean-field Itô-Taylor formula, error estimate. AB -

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.