setting of this paper, [@bertoinlindnermaller08 Thm 3.9] shows in particular, that the exponential functional $V$ as in is absolutely continuous, whenever the subordinator $\eta$ has a strictly positive drift. Further, in [@kuznetsovetal Cor. 2.5], it is shown that the exponential functional $V$ as in is absolutely continuous with continuous density if $\sigma_\xi>0$.\
Nevertheless, if $\eta$ and $\xi$ both are compound Poisson processes, examples can be constructed in which $V$ is not absolutely continuous (see [@lindnersato09] and Remark \[rem:lindnersato\] below).\
The following theorem provides an integro-differential equation fulfilled by the density of $V$ whenever it exists. Notice that for the special case of a deterministic process $\eta_t=t$ this result has been obtained in [@CarmonaPetitYor] using a different technique. In particular, case (ii) below is a special case of the results in [@CarmonaPetitYor] or similarly of [@PardoRiveroSchaik Thm. 2.3] and is just kept here for completeness.

\[thm:density\] Assume that $\xi=(\xi_t)_{t\geq 0}$ is a Lévy process such that $\lim_{t\to \infty}\xi_t=\infty$ and with characteristic triplet $(\gamma_\xi, \sigma^2_\xi, \nu_\xi)$ such that $\int_{[-1,1]}|x|\nu_\xi(dx)<\infty$ and set $\gamma_0:= \gamma_\xi - \int_{[-1,1]}x\nu_\xi(dx)$. Let $\eta=(\eta_t)_{t\geq 0}$ be a subordinator with drift $a_\eta$ and jump measure $\nu_\eta$, independent of $\xi$ and such that at least one of the processes $\xi$ and 