Mean-field forward and backward SDEs with jumps. Associated nonlocal quasi-linear integral-PDEs

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn, Li, Peng and Rainer [6] to BSDEs, the existence and the uniqueness of the solution $(Y^{t,\xi}, Z^{t,\xi}, H^{t,\xi})$, $(Y^{t,x,P_\xi}, Z^{t,x,P_\xi}, H^{t,x,P_\xi})$ of the split equations are proved. The first and the second order derivatives of the process $(Y^{t,x,P_\xi}, Z^{t,x,P_\xi}, H^{t,x,P_\xi})$ with respect to $x$, the derivative of the process $(Y^{t,x,P_\xi}, Z^{t,x,P_\xi}, H^{t,x,P_\xi})$ with respect to the measure $P_\xi$, and the derivative of the process $(\partial_\mu Y^{t,x,P_\xi}(y), \partial_\mu Z^{t,x,P_\xi}(y), \partial_\mu H^{t,x,P_\xi}(y))$ with respect to $y$ are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in $L^2$. The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function $V(t,x,P_\xi):=Y_t^{t,x,P_\xi}$ is regular and allows to show with the help of a new It\^{o} formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type.