A potential-theoretic approach to optimal stopping in a spectrally L\'evy Model

TL;DR

This paper introduces a potential-theoretic method for solving optimal stopping problems involving spectrally negative Lévy processes, enabling direct, constructive solutions even for complex continuation regions.

Contribution

It develops a systematic, potential theory-based approach for optimal stopping in spectrally negative Lévy models, allowing for general, constructive solutions without pre-specified forms.

Findings

Derived necessary and sufficient optimality conditions.

Provided a step-by-step solution procedure for complex problems.

Applicable to a broad class of reward functions.

Abstract

We establish a systematic solution method for optimal stopping problems of spectrally negative L\'evy processes. Our approach relies essentially on the potential theory, in particular the Riesz decomposition and the maximum principle. Using these mathematical results, we not only derive necessary and sufficient conditions of optimality for a broad class of reward functions, but also develop a method to tackle general problems in a direct and constructive way (without pre-specifying the solution form). To reinforce the latter point, we provide a step-by-step solution procedure applicable to complex solution structures, including continuation regions with multiple connected components.