A decomposition for Lévy processes inspected at Poisson moments. (15th June 2023) - Boxma, Onno; Mandjes, Michel
Abstract: 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 T_β with parameter β . The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to T_β, denoted by Y_β, ω . Our main result is a decomposition: we derive a remarkable distributional equality that contains Y_β, ω as well as the running maximum process Y(t) at the exponentially distributed times T_β and T_β+ω . Concretely, Y(T_β) can be written as the sum of two independent random variables that are distributed as Y_β, ω and Y(T_β+ω) . The distribution of Y_β, 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 Cramér–Lundberg insurance risk model.