fulfilling $$\begin{aligned}
\label{eq:densityCOGARCH}
(\beta - \eta  t )f(t)  + \int_{\frac{\beta}{\eta}}^t \nu_S\left(\left(\frac{t-s}{s\varphi}, \infty\right)\right) f(s) ds=0, \quad t\geq \frac{\beta}{\eta}.\end{aligned}$$ Now, if for example $(S_t)_{t\geq 0}$ is chosen to be a Poisson process with intensity $c>0$, we obtain from the following difference-differential equation for the cumulative distribution function $F(t)$ of $V$ $$\frac{\eta t - \beta}{c} F'(t)=F(t)-F(\frac{\beta}{\eta}), \quad t\geq \frac{\beta}{\eta},$$ with $F(t)=0$ for $t< \frac{\beta}{\eta}$. Similarly, for the common choice of $L$ having standard normally distributed jumps, one derives the recursive formula $$f(t) =\frac{2}{\beta-\eta t} \int_{\frac{\beta}{\eta}}^t \left( 1- \phi\left(\sqrt{\frac{t-s}{s\varphi}} \right)\right) f(s) ds, \quad t> \frac{\beta}{\eta},$$ where $\phi$ is the cumulative distribution function of the normal distribution.

(Semi-)Selfdecomposability and Generalized Gamma Convolutions {#sec:prelim}
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We will use the following notations for the classes of infinitely divisible distributions: $$\begin{aligned}
{\mathrm{ID}}, {\mathrm{ID}}^+  \quad & \quad \text{infinitely divisible distributions on }{\mathbb{R}}, {\mathbb{R}}_+ \text{ (respectively)} \\
{\mathrm{ID}}_{\log}, {\mathrm{ID}}^+_{\log} \quad & \quad  \text{infinitely divisible distributions on }{\mathbb{R}}, {\mathbb{R}}_+ \text{ with finite log-moment} \end{aligned}$$ Further the following classes of distributions will be introduced in the next subsections: $$\begin{aligned}
{\mathrm{L}}, {\mathrm{L}}^+ \quad & \quad  \text{selfdecomposable distributions on }{\mathbb{R}}, {\mathbb{R}}_+\\
{\mathrm{L}}(c), {\mathrm{L}}(c)^+ \quad & \quad  c\text{-decomposable/ semi-selfdecomposable distributions on }{\mathbb{R}}, {\mathbb{R}}_+\\
{\mathrm{BO}}\quad & \quad  \text{Goldie-Steutel-Bondesson class, Bondesson's class (on } {\mathbb{R}}_+)\\
{\mathrm{T}}\quad & 