Abstract
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 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 \(A(t):= E[Y(t+\delta ) | \mathscr {F}_t]\). 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 insider influenced asset price process. (ii) Optimal consumption rate from a cash flow modeled as a geometric Ito-Levy SDE, with respect to predictive recursive utility.
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