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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.

##### MSC:
 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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