The stochastic maximum principle for a singular control problem

Abstract

We consider a stochastic control problem in which the control has two components: the first being absolutely continuous, and the second singular. We assume linear dynamics, convex cost criterion and convex state constraint, and we allow the coefficients of the system to be random and the absolutely continuous component of the control to enter both the drift and diffusion coefficients. We do not impose any Lp -bounds on the control. We obtain for this model a Stochastic Maximum Principle in integral form. This is the first version of the Stochastic Maximum Principle that covers the Stochastic Singular Control Problem. When we assume, as in other versions of the Stochastic Maximum Principle, that the admissible controls are square-integrable, we obtain not only a necessary but also a sufficient condition for optimality. The mathematical tools are those of Stochastic Calculus and Convex Analysis.