Existence of optimal controls for singular control problems with state constraints

Abstract

1. Introduction. This paper is concerned with a class of singular control problems with state constraints. The presence of state constraints, a key feature of the problem, refers to the requirement that the controlled diffusion process take values in a closed convex cone at all times [see (3)]. We consider an infinite horizon discounted cost of the form (4). The main objective of the paper is to establish the existence of an optimal control. Singular control is a well-studied but rather challenging class of stochastic control problems. We refer the reader to [7], especially the sections at the end of each chapter, for a thorough survey of the literature. Classical compactness arguments that are used for establishing the existence of optimal controls for problems with absolutely continuous control terms (cf. [8]) do not naturally extend to singular control problems. For one-dimensional models, one can typically establish existence constructively, by characterizing an optimally controlled process as a reflected diffusion (cf. [2, 3, 15]). In higher dimensions, one approach is to study the regularity of solutions of variational inequalities associated with singular control problems and the smoothness of the corresponding free boundary. Such smoothness results are the starting points in the characterization of the optimally controlled process as a constrained diffusion with reflection at the free boundary. Excepting specific models (cf. [30, 31]), this approach encounters substantial difficulties, even for linear dynamics (cf. [32]); a key difficulty is that little is known about the regularity of the free boundary in higher dimensions. Alternative approaches for establishing the existence of optimal controls based on compactness arguments are developed in [12, 17, 25]. The first of these papers considers linear dynamics, while the last two consider models with nonlinear coefficients. In all