Implicit renewal theory for exponential functionals of L\'evy processes

TL;DR

This paper develops a new integral equation and a three-term factorization for the probability density of exponential functionals of Lévy processes, enhancing understanding of their distributional properties and tail behaviors.

Contribution

It introduces a novel integral equation and a Wiener-Hopf type factorization for the law of exponential functionals of Lévy processes, providing new insights and simpler proofs for existing results.

Findings

Derived a new integral equation for the density

Established a three-term Wiener-Hopf type factorization

Analyzed tail and local behavior of the distribution

Abstract

We establish a new integral equation for the probability density of the exponential functional of a L\'evy process and provide a three-term (Wiener-Hopf type) factorisation of its law. We explain how these results complement the techniques used in the study of exponential functionals and, in some cases, provide quick proofs of known results and derive new ones. We explain how the factors appearing in the three-term factorisation determine the local and asymptotic behaviour of the law of the exponential functional. We describe the behaviour of the tail distribution at infinity and of the distribution at zero under some mild assumptions.