A stochastic maximum principle of mean-field type with monotonicity conditions

The objective of this paper is to weaken the Lipschitz condition to a monotonicity condition and to study the corresponding Pontryagin stochastic maximum principle (SMP) for a mean-field optimal control problem under monotonicity conditions.The dynamics of the controlled state process is governed by a mean-field stochastic differential equation (SDE) whose coefficients depend not only on the control, the controlled state process itself but also on its law, and in particular, these coefficients satisfy the monotonicity condition with respect to both the controlled state process and its distribution. The associated cost functional is also of mean-field type. Under the assumption of a convex control domain we derive the SMP, which provides a necessary optimality condition for control processes. Under additional convexity assumptions on the Hamiltonian, we further prove that this necessary condition is also a sufficient one. To achieve this, we first address the challenges related to the existence and the uniqueness of solutions for mean-field backward stochastic differential equations and mean-field SDEs whose coefficients satisfy monotonicity conditions with respect to both the solution as well as its distribution. On the other hand we also construct several illustrative examples demonstrating the generality of our results compared to existing literature.