Stochastic Maximum Principle for Partial Information Optimal Control Problem of Forward‐Backward Systems Involving Classical and Impulse Controls

We study the partial information classical and impulse controls problem of forward‐backward systems driven by Lévy processes, where the control variable consists of two components: the classical stochastic control and the impulse control; the information available to the controller is possibly less than the full information, that is, partial information. We derive a maximum principle to give the sufficient and necessary optimality conditions for the local critical points of the classical and impulse controls problem. As an application, we apply the maximum principle to a portfolio optimization problem with piecewise consumption processes and give its explicit solutions.