On the density of exponential functionals of Lévy processes

Abstract

In this paper, we study the existence of the density associated with the exponential functional of the Lévy process $\xi$,

\[I_{\mathbf{e} _{q}}:=\int_{0}^{\mathbf{e} _{q}}\mathrm{e}^{\xi_{s}}\,\mathrm{d}s,\]

where $\mathbf{e} _{q}$ is an independent exponential r.v. with parameter $q\geq0$. In the case where $\xi$ is the negative of a subordinator, we prove that the density of $I_{\mathbf{e}_{q}}$, here denoted by $k$, satisfies an integral equation that generalizes that reported by Carmona et al. [7]. Finally, when $q=0$, we describe explicitly the asymptotic behavior at $0$ of the density $k$ when $\xi$ is the negative of a subordinator and at $\infty$ when $\xi$ is a spectrally positive Lévy process that drifts to $+\infty$.