Mean-field BDSDEs and associated nonlocal semi-linear backward stochastic partial differential equations

Abstract: In this paper we investigate mean-field backward doubly stochastic differential equations (BDSDEs), i.e., BDSDEs whose driving coefficients also depend on the joint law of the solution process as well as the solution of an associated mean-field forward SDE. Unlike the pioneering paper on BDSDEs by Pardoux-Peng (1994), we handle a driving coefficient in the backward integral of the BDSDE for which the Lipschitz assumption w.r.t. the law of the solution is sufficient, without assuming that this Lipschitz constant is small enough. On the other hand, as the parameters $(x,P_\xi)$ and $(x,P_\xi,y)$ run an infinite-dimensional space, unlike Pardoux and Peng, we cannot apply Kolmogorov's continuity criterion to the value function $V(t,x,P_{\xi}):=Y_t^{t,x,P_{\xi}}$, while in the classical case studied in Pardoux-Peng the value function $V(t,x)=Y_t^{t,x}$ can be shown to be of class $C^{1,2}([0,T]\times\mathbb{R}^d)$, we have for our value function $V(t,x,P_{\xi})$ and its derivative $\partial_\mu V(t,x,P_{\xi},y)$ only the $L^2$-differentiability with respect to $x$ and $y$, respectively. Using a new method we prove the characterization of $V=(V(t,x,P_{\xi}))$ as the unique solution of the associated mean-field backward stochastic PDE.