An optimal stopping problem for spectrally negative Markov additive processes

TL;DR

This paper extends optimal stopping theory to spectrally negative Markov additive processes, deriving differential equations for stopping boundaries using scale matrices, and applies the results to classical problems like Shepp-Shiryaev.

Contribution

It introduces a framework for optimal stopping in Markov additive processes, replacing scale functions with scale matrices, and characterizes boundaries via differential equations.

Findings

Derived differential equations for stopping boundaries in Markov additive processes.

Extended classical optimal stopping problems to the Markov additive setting.

Demonstrated applications to Shepp-Shiryaev and capped maximum problems.

Abstract

Previous authors have considered optimal stopping problems driven by the running maximum of a spectrally negative L\'evy process $X$, as well as of a one-dimensional diffusion. Many of the aforementioned results are either implicitly or explicitly dependent on Peskir's maximality principle. In this article, we are interested in understanding how some of the main ideas from these previous works can be brought into the setting of problems driven by the maximum of a class of Markov additive processes (more precisely Markov modulated L\'evy processes). Similarly to previous works in the L\'evy setting, the optimal stopping boundary is characterised by a system of ordinary first-order differential equations, one for each state of the modulating component of the Markov additive process. Moreover, whereas scale functions played an important role in the previously mentioned work, we work instead with scale matrices for Markov additive processes here. We exemplify our calculations in the setting of the Shepp-Shiryaev optimal stopping problem, as well as a family of capped maximum optimal stopping problems.