  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 $Y(t), Z(t),
K(t,\cdot)$. The driver of the BSDE at time $t$ 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}.
