Optimal control of predictive mean-field equations and applications to finance

We study a coupled system of controlled stochastic differential equations (SDEs) driven by a Brownian motion and a compensated Poisson random measure, consisting of a forward SDE in the unknown process $X(t)$ and a \emph{predictive mean-field} backward SDE (BSDE) in the unknowns $Y(t), Z(t), K(t,\cdot)$\mathrm{.}ThedriveroftheBSDEattime$t$ may depend not just upon the unknown processes $Y(t),Z(t),K(t,\cdot )$, but also on the predicted future value $Y(t+\delta )$, defined by the conditional expectation $A(t):= E[Y(t+\delta) | $\mathcal{F}$_t]$. \ We give a sufficient and a necessary maximum principle for the optimal control of such systems, and then we apply these results to the following two problems:\ (i) Optimal portfolio in a financial market with an \emph{insider influenced asset price process.} \ (ii) Optimal consumption rate from a cash flow modeled as a geometric It^ o-L' evy SDE, with respect to \emph{predictive recursive utility}.