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abstract: 'We study the exponential functional $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ of two one-dimensional independent Lévy 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 Lévy 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.'
author:
- 'Anita Behme[^1]'
title: Exponential functionals of Lévy processes with jumps
---

2010 [*Mathematics subject classification.*]{} 60G10, 60G51, 60E07.\
[*Key words and phrases.*]{} COGARCH volatility, exponential functional, generalized gamma convolution, generalized Ornstein-Uhlenbeck process, integral mapping, Lévy process, selfdecomposability, stationarity

Introduction
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Given two independent Lévy processes $(\xi_t)_{t\geq 0}$, $(\eta_t)_{t\geq 0}$ the corresponding [*exponential functional*]{} is defined as $$\label{eq:expfunc}
 V:=\int_{(0,\infty)} e^{-\xi_{t-}}d\eta_t,$$ provided that the integral converges a.s. Necessary and sufficient conditions for this convergence in terms of the Lévy characteristics of $(\xi_t)_{t\geq 0}$ and $(\eta_t)_{t\geq 