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

Abstract

Equivalences are known between problems of singular stochastic control (SSC) with convex performance criteria and related questions of optimal stopping, see for example Karatzas and Shreve [SIAM J. Control Optim. 22 (1984)]. The aim of this paper is to investigate how far connections of this type generalise to a non convex problem of purchasing electricity. Where the classical equivalence breaks down we provide alternative connections to optimal stopping problems. 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 their (optimal) boundaries. Unlike the case of convex SSC problems we find that the optimal boundaries may be both reflecting and repelling and it is natural to interpret the problem as one of SSC with discretionary stopping.

arXiv comment24 pages, 2 figures

Cited by in corpus (6)

• A solvable two-dimensional singular stochastic control problem with non convex costs
• Nonlinear Filtering of Partially Observed Systems arising in Singular Stochastic Optimal Control
• Expected Supremum Representation of the Value of a Singular Stochastic Control Problem
• Expected Supremum Representation of a Class of Single Boundary Stopping Problems
• On optimal control of reflected diffusions
• Optimal Entry to an Irreversible Investment Plan with Non Convex Costs