Mean-field stochastic differential equations and associated PDEs

Abstract

In this paper, we consider a mean-field stochastic differential equation, also called the McKean–Vlasov equation, with initial data $(t,x)\in[0,T]\times\mathbb{R}^{d}$, whose coefficients depend on both the solution $X^{t,x}_{s}$ and its law. By considering square integrable random variables $\xi$ as initial condition for this equation, we can easily show the flow property of the solution $X^{t,\xi}_{s}$ of this new equation. Associating it with a process $X^{t,x,P_{\xi}}_{s}$ which coincides with $X^{t,\xi}_{s}$, when one substitutes $\xi$ for $x$, but which has the advantage to depend on $\xi$ only through its law $P_{\xi}$, we characterize the function $V(t,x,P_{\xi})=E[\Phi(X^{t,x,P_{\xi}}_{T},P_{X^{t,\xi}_{T}})]$ 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.