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Mean-field stochastic differential equations and associated PDEs. (English) Zbl 1402.60070
Summary: 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 P. L. Lions in [“Théorie des jeu à champs moyens”, Course at Collège de France, 2013], and we extend it in a direct way to the second-order derivatives.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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