Weak solutions of mean-field stochastic differential equations

 报告题目： Weak solutions of mean-field stochastic differential equations 报 告 人： 李娟 教授(山东大学) 报告时间： 2018年05月21日 16:30--17:30 报告地点： 理学院东北楼四楼报告厅（404） 报告摘要： In this talk we discuss weak solutions of mean-field stochastic differential equations (SDEs), also known as McKean-Vlasov equations, whose drift $b(s, X_s,Q_{X_s})$, and diffusion coefficient $sigma(s, X_s,Q_{X_s})$ depend not only on the state process $X_s$ but also on its law. We suppose that $b$ and $sigma$ are bounded and continuous in the state as well as the probability law; the continuity with respect to the probability law is understood in the sense of the 2-Wasserstein metric. Using the approach through a local martingale problem, we prove the existence and the uniqueness in law of the weak solution of mean-field SDEs. The uniqueness in law is obtained if the associated Cauchy problem possesses for all initial condition $fin C_0^infty({mathbb R}^d)$ a classical solution. However, unlike the classical case, the Cauchy problem is a mean-field PDE as recently studied by Buckdahn, Li, Peng and Rainer (2014). In our approach, we also extend the It^o formula associated with mean-field problems given by Buckdahn, Li, Peng and Rainer (2014) to a more general case of coefficients.