Mean-field stochastic differential equations and associated PDEs

In this paper, we consider a mean-field stochastic differential equation, also called the McKean.Vlasov equation, with initial data (t,x).[0,T].Rd, whose coefficients depend on both the solution Xt,xs and its law. By considering square integrable random variables . as initial condition for this equation, we can easily show the flow property of the solution Xt,.s of this new equation. Associating it with a process Xt,x,P.s which coincides with Xt,.s, when one substitutes . for x, but which has the advantage to depend on . only through its law P., we characterize the function V(t,x,P.)=E[.(Xt,x,P.T,PXt,.T)] under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a nonlocal partial differential equation of mean-field type, involving the first- and the second-order derivatives of V with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first- and second-order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding Itô formula. In our approach, we use the notion of derivative with respect to a probability measure with finite second moment, introduced by Lions in [Cours au Collège de France: Théorie des jeu à champs moyens (2013)], and we extend it in a direct way to the second-order derivatives.