Generalized scale functions for spectrally negative L\'evy processes

TL;DR

This paper extends the concept of scale functions for spectrally negative Lévy processes to more general functionals involving the process and its supremum, providing new analytical tools for exit problems.

Contribution

Introduction of generalized scale functions expressed via excursion theory, extending classical scale functions to handle more complex functionals of spectrally negative Lévy processes.

Findings

Derived explicit formulas for generalized scale functions

Extended classical scale functions to broader functionals

Provided new analytical methods for exit problems

Abstract

For a spectrally negative L\'evy process, scale functions appear in the solution of two-sided exit problems, and in particular in relation with the Laplace transform of the first time it exits a closed interval. In this paper, we consider the Laplace transform of more general functionals, which can depend simultaneously on the values of the process and its supremum up to the exit time. These quantities will be expressed in terms of generalized scale functions, which can be defined using excursion theory. In the case the functional does not depend on the supremum, these scale functions coincide with the ones found on the literature, and therefore the results in this work are an extension of them.