Doubly-stochastic interpretation for nonlocal semi-linear backward stochastic partial differential equations

Abstract

Motivated by P.L. Lions' seminal course on Mean Field Games (MFG) and the associated Master equation at Collège de France [19], we investigate in the case of absence of control mean-field backward doubly stochastic differential equations (BDSDEs), i.e., BDSDEs whose driving coefficients also depend on the joint law of the solution process. This BDSDE can be interpreted as randomly perturbed recursive cost functional which we associate with a forward mean-field SDE. Defining the value function V=V(t,x,μ) through this BDSDE, we prove that V=V(t,x,μ) is the unique classical solution of a backward stochastic PDE over [0,T]×Rd×P2(Rd). This backward SPDE extends in the non-control case the Master equation studied in earlier works by different authors. The associated MFG can be interpreted as a forward dynamics endowed with a randomly perturbed recursive cost functional, the solution of the MFG system (u(t,x),m(t)) remains also in this extended case related with V=V(t,x,μ) by the relation u(t,x)=V(t,x,m(t)).Our main objective is to characterise the random field V as classical solution of the backward SPDE. For this we study the regularity of V over [0,T]×Rd×P2(Rd). However, unlike the pioneering paper on BDSDEs by Pardoux-Peng [25] where V does not depend on the measure, here we have only an L2-regularity of V with respect to its variables. Other difficulties to overcome and subtle technical proofs are related with the study of the L2-regularity and, hence, that of the solution of the underlying mean-field BDSDE. It adds that our BDSDE needs weaker assumptions on the coefficients than in [25]. Finally, let us point out that for our approach we extend the classical mean-field Itô formula to smooth functions of solutions of mean-field BDSDEs. This formula contains earlier (mean-field) Itô formulas as special cases.