u \psi_\mu'(u) + \frac{\sigma^2_\xi}{2} u^2\left( (\psi'_\mu(u))^2 - \psi''_\mu(u) \right) \\
 & + \int_{{\mathbb{R}}} \left( e^{\psi_\mu(u) - \psi_\mu(ue^{-y})} - 1 - u \psi'_\mu(u) y \mathds{1}_{|y|\leq 1}\right) \nu_\xi(dy), \quad u >0. \nonumber\end{aligned}$$ Starting from this, we will consider several aspects of exponential functionals. In particular, in Section \[sec:density\], we derive an integro-differential equation for the density of the exponential functional (given its existence) which extends a previous result from [@CarmonaPetitYor] where $\eta$ was assumed to be deterministic.\
Since selfdecomposable distributions and generalized Gamma convolutions play an important role in the remainder of the paper, we review them and their connection to exponential functionals in Section \[sec:prelim\], which also includes some new results on these classes of distributions. Further, Section \[sec:nested\] is concerned with the behaviour of the class of distributions of exponential functionals for varying characteristics of $\xi$. In Sections \[sec:selfdec\] and \[sec:ggc\] we derive general conditions for selfdecomposable distributions to be given by an exponential functional with predetermined process $\xi$ and also apply these on generalized Gamma convolutions. Finally, Section \[sec:proof\] contains the proof of Proposition \[prop:GGCfactorinBO\].

Notation {#notation .unnumbered}
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We write $\mu={\mathcal{L}}(X)$ if $\mu$ is the distribution of the random variable $X$. The set of all probability distributions on 