A decomposition for Lévy processes inspected at Poisson moments

Abstract We consider a Lévy process Y ( t ) that is not continuously 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 distribution running maximum such inspection up to , denoted by $Y_{\beta,\omega}$ Our main result decomposition: we derive remarkable distributional equality contains as well $\bar Y(t)$ times and $T_{\beta+\omega}$ Concretely, $\overline{Y}(T_\beta)$ can be written sum two independent random variables are $\overline{Y}(T_{\beta+\omega})$ identified more explicitly in special cases spectrally positive negative process. As illustrative example potential our results, show how determine asymptotic behavior bankruptcy probability Cramér–Lundberg insurance risk model.