A Non Convex Singular Stochastic Control Problem and Its Related Optimal Stopping Boundaries

Abstract

We show that the equivalence between certain problems of singular stochastic control (SSC) and related questions of optimal stopping known for convex performance criteria (see, for example, Karatzas and Shreve (1984)) continues to hold in a non convex problem provided a related discretionary stopping time is introduced. Our problem is one of storage and consumption for electricity, a partially storable commodity with both positive and negative prices in some markets, and has similarities to the finite fuel monotone follower problem. In particular we consider a non convex infinite time horizon SSC problem whose state consists of an uncontrolled diffusion representing a real-valued commodity price, and a controlled increasing bounded process representing an inventory. We analyse the geometry of the action and inaction regions by characterising the related optimal stopping boundaries.

Keywords: finite-fuel singular stochastic control, optimal stopping, free-boundary, smooth-fit, Hamilton-Jacobi-Bellman equation, irreversible investment

JEL Classification: C02, C61, E22, D92