A decomposition for Levy processes inspected at Poisson moments

Abstract

We consider a Lévy 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 Lévy 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 Cramér-Lundberg insurance risk model.