A maximum principle for progressive optimal control of mean‐field forward–backward stochastic system involving random jumps and impulse controls

Abstract

In this paper, we study an optimal control problem of a mean‐field forward–backward stochastic system with random jumps in progressive structure, where both regular and singular controls are considered in our formula. In virtue of the variational technology, the related stochastic maximum principle (SMP) has been obtained, and it is essentially different from that in the classical predictable structure. Specifically, there are three parts in our SMP, that is, continuous part, jump part, and impulse part, and they are, respectively, used to characterize the characteristics of the optimal controls at continuous time, jump time, and impulse time. This shows that the progressive structure can more accurately describe the characteristics of the optimal control at the jump time. We also give two linear–quadratic examples to show the significance of our results.