  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.
