Optimal stopping of Hunt and L\'evy processes

TL;DR

This paper develops a spectral representation approach for optimal stopping problems of Hunt and Lévy processes, providing explicit solutions and applications to Brownian motion with jumps and Poisson processes.

Contribution

It introduces a spectral representation of the value function for infinite horizon optimal stopping problems of Hunt processes, especially Lévy processes, using Green kernels and Wiener-Hopf factorization.

Findings

Spectral representation of the value function in terms of Green kernels.

Explicit solutions for Brownian motion with exponential jumps.

Application to Poisson processes with exponential jumps and negative drift.

Abstract

The optimal stopping problem for a Hunt processes on $\R$ is considered via the representation theory of excessive functions. In particular, we focus on infinite horizon (or perpetual) problems with one-sided structure, that is, there exists a point $x^*$ such that the stopping region is of the form $[x^*,+\infty)$. Corresponding results for two-sided problems are also indicated. The main result is a spectral representation of the value function in terms of the Green kernel of the process. Specializing in L\'evy processes, we obtain, by applying the Wiener-Hopf factorization, a general representation of the value function in terms of the maximum of the L\'evy process. To illustrate the results, an explicit expression for the Green kernel of Brownian motion with exponential jumps is computed and some optimal stopping problems for Poisson process with positive exponential jumps and negative drift are solved.