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

TL;DR

This paper introduces a novel 'one-shot' solution method for a finite-fuel singular control problem with discretionary stopping, addressing an open question and revealing complex behaviors of the solution in limited fuel scenarios.

Contribution

It presents a new 'one-shot' approach to solve an open problem in finite-fuel control, showing that the waiting region may be disconnected and that solutions differ even with small fuel amounts.

Findings

Waiting region can be disconnected in the solution.

Solution without fuel does not approximate small-fuel solutions.

Recommended solving the 'one-shot' problem before traditional methods.

Abstract

We introduce a novel 'one-shot' solution technique resolving an open problem (Karatzas et al., Finite-fuel singular control with discretionary stopping, Stochastics 71:1-2 (2000)). Unexpectedly given the convexity of the latter problem, its waiting region is not necessarily connected. Along a typical sample path, the state process may even spend positive time in both of its connected components. The analysis reveals more generally that when fuel is limited, contrary to intuition, the solution without fuel is not necessarily indicative of the solution for small amounts of fuel. To resolve this, we recommend solving the `one-shot' problem, which is one of optimal stopping, prior to employing the usual `guess and verify' solution approach.