Two-sided optimal stopping for L\'evy processes

TL;DR

This paper develops a comprehensive framework for solving infinite horizon two-sided optimal stopping problems for Lévy processes, providing verification theorems, threshold angle calculations, and explicit solutions for specific jump processes.

Contribution

It introduces a two-sided verification theorem based on supremum and infimum, and computes the value function's angle at optimal thresholds, extending the theory beyond smooth-pasting cases.

Findings

Verification theorem in terms of supremum and infimum

Explicit solution for compound Poisson process with exponential jumps

Threshold angle computation at optimal stopping boundaries

Abstract

Infinite horizon optimal stopping problems for a L\'evy processes with a two-sided reward function are considered. A two-sided verification theorem is presented in terms of the overall supremum and the overall infimum of the process. A result to compute the angle of the value function at the optimal thresholds of the stopping region is given. To illustrate the results, the optimal stopping problem of a compound Poisson process with two-sided exponential jumps and a two-sided payoff function is solved. In this example, the smooth-pasting condition does not hold.