A general stochastic maximum principle for mixed relaxed-singular control problems

TL;DR

This paper develops a Pontryagin maximum principle for optimal control problems involving both measure-valued and singular controls in stochastic systems, broadening the scope of stochastic control theory.

Contribution

It introduces necessary optimality conditions for mixed relaxed-singular controls in non-convex stochastic systems, extending existing maximum principles.

Findings

Established a stochastic maximum principle for mixed relaxed-singular controls.

Proved existence of optimal controls under general conditions.

Extended the theory to non-convex control domains.

Abstract

We consider in this paper, mixed relaxed-singular stochastic control problems, where the control variable has two components, the first being measure-valued and the second singular. The control domain is not necessarily convex and the system is governed by a nonlinear stochastic differential equation, in which the measure-valued part of the control enters both the drift and the diffusion coefficients. We establish necessary optimality conditions, of the Pontryagin maximum principle type, satisfied by an optimal relaxed-singular control, which exist under general conditions on the coefficients. The proof is based on the strict singular stochastic maximum principle established by Bahlali-Mezerdi, Ekeland's variational principle and some stability properties of the trajectories and adjoint processes with respect to the control variable.