A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes

TL;DR

This paper establishes a Wiener-Hopf type factorization for the exponential functional of Lévy processes, providing new distributional representations and properties, and introduces a refined Markovian approach for analyzing related Ornstein-Uhlenbeck processes.

Contribution

It introduces a novel Wiener-Hopf type factorization for exponential functionals of Lévy processes, along with new distributional formulas and a refined Markovian analysis method.

Findings

Factorization of exponential functional into independent components.

Integral and power series representations for the law of the functional.

New distributional properties and insights into Lévy processes with jumps.

Abstract

For a L\'evy process $\xi=(\xi_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows \[{\rm{I}}_{\xi}=\int_0^{\infty}e^{\xi_t} dt.\] Under mild conditions on $\xi$, we show that the following factorization of exponential functionals \[{\rm{I}}_{\xi}\stackrel{d}={\rm{I}}_{H^-} \times {\rm{I}}_{Y}\] holds, where, $\times $ stands for the product of independent random variables, $H^-$ is the descending ladder height process of $\xi$ and $Y$ is a spectrally positive L\'evy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of ${\rm{I}}_{\xi}$ for a large class of L\'evy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein-Uhlenbeck processes which is of independent interest on its own. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process.