Dynkin Games for L\'evy Processes

TL;DR

This paper develops a verification theorem for Dynkin games driven by Le9vy processes, providing a method to determine the value function and optimal stopping rules using averaging functions and fluctuation identities.

Contribution

It introduces a novel verification approach for Le9vy-driven Dynkin games, linking the value function to averaging functions and fluctuation identities.

Findings

Derived a verification theorem for Le9vy process-based Dynkin games.

Identified optimal stopping times as hitting times of support sets.

Presented three examples illustrating the method and properties of solutions.

Abstract

We obtain a verification theorem for solving a Dynkin game driven by a L\'evy process. The result requires finding two averaging functions that, composed respectively with the supremum and the infimum of the process, summed, and taked the expectation, provide the value function of the game. The optimal stopping rules are the respective hitting times of the support sets of the averaging functions. The proof relies on fluctuation identities of the underlying L\'evy process. We illustrate our result with three new simple examples, where the smooth pasting property of the solutions is not always present.