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 $\xi$, $$I_{\ee_q}:=\int_0^{\ee_q} e^{\xi_s} \, \mathrm{d}s,$$ where $\ee_q$ is an independent exponential r.v. with parameter $q\geq 0$. In the case when $\xi$ is the negative of a subordinator, we prove that the density of $I_{\ee_q}$, 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 $\infty$ when $\xi$ is a spectrally positive L\'evy process that drifts to $+\infty$.