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Implicit renewal theory for exponential functionals of Lévy processes

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Published
<mark>Journal publication date</mark>30/09/2023
<mark>Journal</mark>Stochastic Processes and their Applications
Volume163
Number of pages26
Pages (from-to)262-287
Publication StatusPublished
Early online date30/06/23
<mark>Original language</mark>English

Abstract

We establish a new integral equation for the probability density of the exponential functional of a Lévy 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.