Optimal stopping, Appell polynomials and Wiener-Hopf factorization representations of excessive functions of L\'evy processes

TL;DR

This paper investigates the optimal stopping problem for Lévy processes, utilizing Appell polynomials and Wiener-Hopf factorization to characterize the value function and its representing measure.

Contribution

It introduces a novel approach linking Appell polynomials and Wiener-Hopf factorization to explicitly represent the value function in optimal stopping for Lévy processes.

Findings

Representation of the value function via Appell polynomials

Explicit form of the representing measure for the value function

Application of Wiener-Hopf factorization in the analysis

Abstract

In this paper we study the optimal stopping problem for L\'evy processes studied by Novikov and Shiryayev, Stochastics, 2007 In particular, we are interested in finding the representing measure of the value function. It is seen that that this can be expressed in terms of the Appell polynomials. An important tool in our approach and computations is the Wiener-Hopf factorization.