A geometric answer to an open question of singular control with stopping

TL;DR

This paper addresses an open problem in singular stochastic control with stopping by identifying moving boundaries through geometric analysis, revealing a discontinuity that advances understanding of the control and stopping problem.

Contribution

It introduces a geometric approach to solve an open singular control problem with stopping, providing candidate boundaries in a previously unresolved parameter range.

Findings

Identification of moving boundaries in a complex control problem

Discovery of a discontinuity in the parameter-dependent geometry

Highlighting the role of geometric analysis in stochastic control

Abstract

We solve a problem of singular stochastic control with discretionary stopping, suggested as an interesting open problem by Karatzas, Ocone, Wang and Zervos (2000), by providing suitable candidates for the moving boundaries in an unsolved parameter range. We proceed by identifying an optimal stopping problem with similar variational inequalities and inspecting its parameter-dependent geometry (in a sense going back to Dynkin (1965)), which reveals a discontinuity not previously exploited. We thus highlight the potential importance of this geometric information in both singular control and parameter-dependent optimal stopping.