Singular mean-field optimal control for forward-backward stochastic systems and applications to finance

Abstract

In this paper, we study a class of singular stochastic optimal control problems for systems described by mean-field forward-backward stochastic differential equations, in which the coefficient depend not only on the state process but also its marginal law of the state process through its expected value. Moreover, the cost functional is also of mean-field type. The control variable has two components, the first being absolutely continuous and the second singular control. Necessary conditions for optimal control for this systems in the form of a Pontrygin maximum principle are established by means convex perturbation techniques for both continuous and singular parts. Our stochastic maximum principle differs from the classical one in the sense that here the adjoint equation has a mean-field type. The control domain is assumed to be convex. As an illustration of our results, we consider a mean-variance portfolio selection mixed with a recursive utility functional optimization problem involving singular control. The explicit expression of the optimal portfolio selection strategy is obtained in the state feedback form involving both state process and its marginal distribution, via the solutions of Riccati ordinary differential equations with time-inconsistent solution.