Weak solutions of mean-field stochastic differential equations

In this article, we studyweak solutions of mean-field stochastic differential equations (SDEs), also known as McKean-Vlasov equations, whose drift b(s, X-s, Q(Xs)), and diffusion coefficient sigma (s, X-s, Q(Xs)) 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 f. is an element of C-0(infinity) (R-d) 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 Ito formula associated with mean-field problems given by Buckdahn et al. to a more general case of coefficients.