Exponential functionals of L\'evy processes with jumps

TL;DR

This paper investigates the distribution and properties of exponential functionals of two independent Lévy processes, deriving equations, analyzing the mapping of distributions, and characterizing classes like selfdecomposable and generalized Gamma convolutions.

Contribution

It introduces new integro-differential equations for the density of exponential functionals and characterizes the distribution classes within the Lévy process framework.

Findings

Derived an integro-differential equation for the density of the exponential functional.

Analyzed the mapping from Lévy process characteristics to the distribution of the exponential functional.

Provided new characterizations of selfdecomposable and generalized Gamma convolution distributions.

Abstract

We study the exponential functional $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ of two one-dimensional independent L\'evy processes $\xi$ and $\eta$, where $\eta$ is a subordinator. In particular, we derive an integro-differential equation for the density of the exponential functional whenever it exists. Further, we consider the mapping $\Phi_\xi$ for a fixed L\'evy process $\xi$, which maps the law of $\eta_1$ to the law of the corresponding exponential functional $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s$, and study the behaviour of the range of $\Phi_\xi$ for varying characteristics of $\xi$. Moreover, we derive conditions for selfdecomposable distributions and generalized Gamma convolutions to be in the range. On the way we also obtain new characterizations of these classes of distributions.