A decomposition for Levy processes inspected at Poisson moments

TL;DR

This paper derives a novel decomposition for the distribution of the maximum of a Lévy process observed at Poisson inspection times, with applications to risk models and explicit cases for spectrally positive/negative processes.

Contribution

It introduces a new distributional decomposition relating the maximum at Poisson inspection times to the overall maximum, with explicit results for special Lévy process classes.

Findings

Distributional equality involving maxima at Poisson inspection times

Explicit identification for spectrally positive/negative Lévy processes

Application to asymptotic bankruptcy probability in risk models

Abstract

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.