A decomposition for Lévy processes inspected at Poisson moments

We consider a Lévy process Y(t) that is not continuously observed, but rather inspected at Poisson(Ω) moments only, over an exponentially distributed time Tβ with parameter β. The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to Tβ, denoted by Yβ,Ω. Our main result is a decomposition: we derive a remarkable distributional equality that contains Yβ,Ω as well as the running maximum process PY (t) at the exponentially distributed times Tβ and Tβ+Ω. Concretely, Y(Tβ) can be written as the sum of two independent random variables that are distributed as Yβ,Ω and Y(Tβ+Ω). The distribution of Yβ,Ω 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 Cramer.Lundberg insurance risk model.