I am reading about Levy processes and am having trouble to understand the following:
Let $X$ be a subordinated Levy Process on ($\Omega$, $\mathcal{F}$, $\mathbb{P}$), and let $\tau$ = $\tau\left(q\right)$ be exponentially distributed with $q>0$.
My question is, why is it that
$\mathbb{E}\left[X_{\tau}\right] = \int_{0}^\infty qe^{-qt} \mathbb{E}\left[X_{t}\right] dt$
And likewise,
$\mathbb{P}\left[X_{\tau} \in dy \right] = \int_{0}^\infty qe^{-qt} \mathbb{P}\left[X_{t} \in dy\right] dt$
I do understand the intuition behind it, however I cannot seem to work out the mathematics behind it starting from the definitions of expectations and probability distributions as integrals, as my $dt$ always disappears.
Any help would be much appreciated.
This is a generic probability question. The idea is that $$ \mathbb{E}[X] = \int_{D(Y)} \mathbb{E}[X|Y=y]f_Y(y)\,\mathrm{d}y $$ where $D(Y)$ is the domain of $Y$. In this case, you take $Y=\tau$ and we have that the density of $f_{\tau}(y) = qe^{-qy}$.