Dynkin Games for Lévy Processes

We obtain a verification theorem for solving a Dynkin game driven by a Lévy process. The result requires finding two averaging functions that, composed, respectively, with the supremum and the infimum of the process, summed, and taking 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. We also provide some hints to understand in which type of problems the verification result can be used. The proof relies on fluctuation identities of the underlying Lévy process. We illustrate our result with examples, three of them with affine payoffs for different processes and a fourth consisting of a mathematical finance application. The smooth pasting property of the solutions is not always present.