The one-shot problem: Solution to an open question of finite-fuel singular control with discretionary stopping

Abstract

We resolve a long-standing open question posed by Karatzas, Ocone, Wang, and Zervos (Stochastics, 2000) on finite-fuel singular stochastic control with discretionary stopping. Our approach introduces a novel "one-shot" solution technique, based on reduction to an auxiliary optimal stopping problem. We prove that, despite the convexity of the objective function, the waiting region of the optimal strategy need not be connected. Furthermore, we identify a qualitative transition in the optimal policy: even with arbitrarily small positive fuel, the solution can differ from the zero-fuel (pure optimal stopping) limit. These results reveal a fundamental non-robustness in such control problems, showing that zero-fuel models need not approximate behaviour under small but positive fuel constraints.