A McKean–Vlasov optimal mixed regular-singular control problem for nonlinear stochastic systems with Poisson jump processes

In this paper, we develop the necessary conditions of optimality for a new class of mixed regular-singular control problem for nonlinear forward–backward stochastic systems with Poisson jump processes of McKean–Vlasov type. The coefficients of the system and the performance functional depend not only on the state process but also its marginal law of the state process through its expected value. The control variable has two components, the first being absolutely continuous and the second singular control. Our optimality conditions for these McKean–Vlasov׳s systems are established by means of convex perturbation techniques for both continuous and singular parts. In our class of McKean–Vlasov control problem, there are two types of jumps for the state processes, the inaccessible ones which come from the Poisson martingale part and the predictable ones which come from the singular control part.