A general maximum principle for mean-field forward-backward doubly stochastic differential equations with jumps processes

Abstract

In this paper, we investigate the optimal control problems for stochastic differential equations (SDEs in short) of mean-field type with jump processes. The control variable is allowed to enter into both diffusion and jump terms. This stochastic maximum principle differs from the classical one in the sense that here the first-order adjoint equation turns out to be a linear mean-field backward SDE with jumps, while the second-order adjoint equation remains the same as in Tang and Li's stochastic maximum principle [32]. Finally, for the reader's convenience we give some analysis results used in this paper in the Appendix.

An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

“…The idea here is to reformulate in the first step the risk-sensitive control problem in terms of an augmented state process and terminal payoff problem. An intermediate stochastic maximum principle (SMP in short) is then obtained by applying the SMP of [1,8] for a loss functional without running cost; for the same particular cases see [7]. Then we transform the intermediate adjoint processes to a simpler form, using the fact that the set of controls is convex.…”