A decomposition for Levy processes inspected at Poisson moments

We consider a L\'evy process $Y(t)$ that is not permanently observed, but rather inspected at Poisson($\omega$) moments only, over an exponentially distributed time $T_\beta$ with parameter $\beta$. The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to $T_\beta$, denoted by $Y_{\beta,\omega}$. Our main result is a decomposition: we derive a remarkable distributional equality that contains $Y_{\beta,\omega}$ as well as the running maximum process $\bar Y(t)$ at the exponentially distributed times $T_\beta$ and $T_{\beta+\omega}$. Concretely, $\overline{Y}(T_\beta)$ can be written the sum of the two independent random variables that are distributed as $Y_{\beta,\omega}$ and $\overline{Y}(T_{\beta+\omega})$. The distribution of $Y_{\beta,\omega}$ can be identified more explicitly in the two special cases of a spectrally positive and a spectrally negative L\'evy process. As an illustrative example of the potential of our results, we show how to determine the asymptotic behavior of the bankruptcy probability in the Cram\'er-Lundberg insurance risk model.