Exponential functionals of Lévy processes with jumps

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.

References in corpus (1)

• On the density of exponential functionals of Lévy processes