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

Abstract

For a Lévy process $\xi =({\xi }_t{)}_{t\ge 0}$ drifting to $-\infty$, we define the so-called exponential functional as follows \[{\rm{I}}_ξ=\int_0^{\infty}e^{ξ_t} dt.\] Under mild conditions on $\xi$, we show that the following factorization of exponential functionals \[{\rm{I}}_ξ\stackrel{d}={\rm{I}}_{H^-} \times {\rm{I}}_{Y}\] 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 Lévy 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 ${\mathrm{I}}_{\xi }$ for a large class of Lévy 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.

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• Recurrent extensions of self-similar Markov processes and Cramér's condition II
• Distributional properties of exponential functionals of Levy processes
• Law of the exponential functional of one-sided Lévy processes and Asian options
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Cited by in corpus (10)

• Asymptotics of self-similar growth-fragmentation processes
• Asymptotics for densities of exponential functionals of subordinators
• A note on a Poissonian functional and a $q$-deformed Dufresne identity
• A temporal factorization at the maximum for spectrally negative positive self-similar Markov processes
• Exponential functionals of Lévy processes with jumps
• Asymptotics for Exponential Functionals of Random Walks
• Moments of exponential functionals of Lévy processes on a deterministic horizon -- identities and explicit expressions
• Asian options and meromorphic Levy processes
• Bivariate Bernstein-gamma functions, potential measures, and asymptotics of exponential functionals of Lévy processes
• Asymptotic Results for Heavy-tailed Lévy Processes and their Exponential Functionals