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

TL;DR

This paper explores the complex relationship between non-convex singular stochastic control problems in electricity purchasing and their connection to optimal stopping, revealing new boundary behaviors and interpretations.

Contribution

It extends the classical equivalence between SSC and optimal stopping to a non-convex setting, analyzing boundary geometries and introducing a discretionary stopping perspective.

Findings

Optimal boundaries can be both reflecting and repelling.

Connections to optimal stopping are generalized beyond classical convex cases.

The problem is interpreted as SSC with discretionary stopping.

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.