is unknown. E.g. Dufresne (cf [@bertoinyor Equation (16)]) showed that $V
\overset{d}= \frac{2}{\sigma^2} G^{-1}_{2a/\sigma^2}$ where $G_k$ is a Gamma($k,1$) random variable, whenever $\xi$ is a Brownian motion with variance $\sigma^2$ and drift $a>0$, and $\eta$ is deterministic. Here and in the following $\overset{d}=$ denotes equality in distribution. A few more concrete distributions of specific exponential functionals have been obtained in [@GjessingPaulsen]. Further it has been investigated whether exponential functionals belong to certain classes of distributions. So, as shown in [@bertoinlindnermaller08], $V$ is selfdecomposable whenever $\xi$ is spectrally negative, i.e. has no positive jumps. In [@BehmeMaejima] conditions are derived under which the exponential functional is a generalized gamma convolution, where one of the processes is a compound Poisson process.

In this article we focus on the case of exponential functionals as in when $\xi$ is a general Lévy process such that $\lim_{t\to \infty}\xi_t=\infty$ and $\eta$ is a subordinator, independent of $\xi$. By [@BLMranges Cor. 1] this means that $V\geq 0$ a.s. and we have the following relationship between the characteristic triplet $(\gamma_\xi, \sigma^2_\xi, \nu_\xi)$ of $\xi$ and the Laplace exponents $\psi_\eta$ and $\psi_\mu$ of $\eta_1$ and the distribution $\mu$ of $V$, resp., $$\begin{aligned}
 \label{eq:relationwithjumps}
 \psi_\eta (u) = & (\gamma_\xi - \frac{\sigma_\xi^2}{2}) 