Maximum Principles for Optimal Control of Forward-Backward Stochastic Differential Equations with Jumps

Abstract

We present various versions of the maximum principle for optimal control of forward-backward stochastic differential equations (SDE) with jumps. Our study is motivated by risk minimization via $g$-expectations. We first prove a general sufficient maximum principle for optimal control with partial information of a stochastic system consisting of a forward and a backward SDE driven by Levy processes. We then present a Malliavin calculus approach which allows us to handle non-Markovian systems. Finally, we give examples of applications.