Optimal stopping of the stable process with state-dependent killing

TL;DR

This paper solves an optimal stopping problem for a stable Lévy process with state-dependent killing, modeling bankruptcy, by characterizing a self-similar Markov process and deriving an explicit stopping rule.

Contribution

It introduces a novel approach to optimal stopping for stable processes with state-dependent killing, maintaining self-similarity and explicitly characterizing the stopping rule.

Findings

Explicit optimal stopping rule derived for the process.

Characterization of a self-similar Markov process associated with the stable process.

Solution applicable to bankruptcy modeling scenarios.

Abstract

We describe the solution of an optimal stopping problem for a stable L\'evy process killed at state-dependent rate, which can be seen as a model for bankruptcy. The killing rate is chosen in such a way that the killed process remains self-similar, and the solution to the optimal stopping problem is obtained by characterising a self-similar Markov process associated with the stable process. The optimal stopping strategy is to stop upon first passage into an interval, found explicitly in terms of the parameters of the model.