On the density of exponential functionals of L\'evy processes

In this paper, we study the existence of the density associated to the exponential functional of the L'evy process $\xi$, $$I_{{\text{ee}}_q}:={\int }_0^{{\text{ee}}_q}{e}^{{\xi }_s}\mathrm{d}s,$$ where ${\text{ee}}_q$ is an independent exponential r.v. with parameter $q\ge 0$. In the case when $\xi$ is the negative of a subordinator, we prove that the density of $I_{{\text{ee}}_q}$, here denoted by $k$, satisfies an integral equation that generalizes the one found by Carmona et al. \cite{Carmona97}. Finally when $q=0$, we describe explicitly the asymptotic behaviour at 0 of the density $k$ when $\xi$ is the negative of a subordinator and at $\mathrm{\infty }$ when $\xi$ is a spectrally positive L'evy process that drifts to $+\mathrm{\infty }$.