Weak solutions of mean-field stochastic differential equations

Abstract

ABSTRACTIn this article, we study weak solutions of mean-field stochastic differential equations (SDEs), also known as McKean–Vlasov equations, whose drift , and diffusion coefficient depend not only on the state process Xs but also on its law. We suppose that b and σ 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 a classical solution. However, unlike the classical case, the Cauchy problem is a mean-field PDE as recently studied by Buckdahn et al. [arXiv:1407.1215, 2014]. In our approach, we also extend the Itô formula associated with mean-field problems given by Buckdahn et al. to a more general case of coefficients.