Fluctuation theory for spectrally negative L\'evy processes killed by additive functionals

TL;DR

This paper extends fluctuation identities for spectrally negative Lévy processes killed by additive functionals, expressing results through generalized scale functions characterized by integral equations.

Contribution

It introduces a unified framework for fluctuation identities involving positive co-natural additive functionals, generalizing classical results with new integral equation characterizations.

Findings

Fluctuation identities retain classical structure with additive functionals.

Generalized scale functions solve Volterra-type integral equations.

Approach uses mixture representations and approximation schemes.

Abstract

In this paper, we study fluctuation identities for spectrally negative L\'evy processes killed by a general class of additive functionals. We consider positive co-natural additive functionals (PcNAFs), which include as special cases both absolutely continuous functionals and finite mixtures of local times. Our main result shows that the associated fluctuation identities, such as two-sided exit problems and resolvent measures, retain the same structure as in the classical case and can be expressed in terms of generalized scale functions. These scale functions are characterized as the unique solutions to Volterra-type integral equations driven by Radon measures, thereby extending the results of Li and Palmowski and Li and Zhou. Our approach is based on representing the additive functional as a mixture of local times with respect to its Revuz measure, combined with classical fluctuation identities and an approximation scheme for general Radon measures using Poisson random measures.