on ${\mathbb{R}}_+$ as well as of generalized Gamma convolutions. This will be done in Corollaries \[cor:densitySD\] and \[cor:densityGGC\] below. For the moment, we end this section with an example of application for Theorem \[thm:density\].

\[ex:COGARCH\] Let $L=(L_t)_{t\geq 0}$ be a Lévy process with characteristic triplet $(\gamma_L,\sigma_L^2,\nu_L)$ and set $$\label{eq-def-S}
 S_t:=[L,L]_t^{d}=\sum_{0<s\leq t} (\Delta L_s)^2, \quad t\geq 0.$$ Then the COGARCH volatility process with parameters $\beta,\eta, \varphi>0$ driven by $L$ or $S$ is defined as $$\label{cog}
V_t = e^{-\xi_t}\left(V_0 + \beta\int_{(0,t]} e^{\xi_{s}} \,d s \right),\quad t\ge0,$$ where $V_0$ is a nonnegative random variable, independent of $(L_t)_{t\geq 0}$, and $$\label{eq-defX}
\xi_t = \eta t - \sum_{0<s\leq t} \log (1+\varphi \Delta S_s),\quad t\geq 0.$$ As originally shown in [@KLM:2004 Thm. 3.1], the process defined in has a strictly stationary distribution if and only if $$\label{cogstatcond} \int_{{\mathbb{R}}_+} \log(1+\varphi y)\,\nu_S(d y) =\int_{{\mathbb{R}}} \log(1+\varphi y^2)\,\nu_L(d y) < \eta$$ and in this case, the stationary distribution is given by the distribution of the exponential functional $$V=\beta \int_{{\mathbb{R}}_+} e^{-\xi_{s}} \,d s.$$ Since $\xi$ is spectrally negative by construction, we can apply Theorem \[thm:density\](i) (or [@CarmonaPetitYor Prop. 2.1]) to obtain that $V$ has a density $f(t)$, $t\geq 0$, with $f(t)=0$ for $t< \frac{\beta}{\eta}$, while $f$ is continuous on $(\frac{\beta}{\eta}, \infty)$ 