Implicit renewal theory for exponential functionals of L\'evy processes
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.
Keywords
Cite
@article{arxiv.1510.01809,
title = {Implicit renewal theory for exponential functionals of L\'evy processes},
author = {Jonas Arista and Víctor M. Rivero},
journal= {arXiv preprint arXiv:1510.01809},
year = {2023}
}
Comments
This version of the paper will appear in Stochastic Processes and their Applications, and replaces an older version. It includes several improvements suggested by the referees in the publication process