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A decomposition for Lévy processes inspected at Poisson moments

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  15 November 2022

Onno Boxma  and
Michel Mandjes
Onno Boxma*
Affiliation:
Eindhoven University of Technology
Michel Mandjes*
Affiliation:
University of Amsterdam
*
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Email: o.j.boxma@tue.nl
**Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands. Email: m.r.h.mandjes@uva.nl
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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 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 as the sum of 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.

Keywords

Lévy processrunning maximumdecompositionbankruptcy probability

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G51: Processes with independent increments; L'evy processes
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 557 - 569
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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