Resumen de Mean-field stochastic differential equations and associated PDEs
Rainer Buckdahn, Juan Li, Shige Peng, Catherine Rainer
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(t,x)∈[0,T]×Rd, whose coefficients depend on both the solution Xt,xsXst,x and its law. By considering square integrable random variables ξξ as initial condition for this equation, we can easily show the flow property of the solution Xt,ξsXst,ξ of this new equation. Associating it with a process Xt,x,PξsXst,x,Pξ which coincides with Xt,ξsXst,ξ, when one substitutes ξξ for xx, but which has the advantage to depend on ξξ only through its law PξPξ, we characterize the function V(t,x,Pξ)=E[Φ(Xt,x,PξT,PXt,ξT)]V(t,x,Pξ)=E[Φ(XTt,x,Pξ,PXTt,ξ)] 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 VV 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.
