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On moments of downward passage times for spectrally negative Lévy processes

Published online by Cambridge University Press:  16 November 2022

Anita Behme*
Affiliation:
Technische Universität Dresden
Philipp Lukas Strietzel*
Affiliation:
Technische Universität Dresden
*
*Postal address: Technische Universität Dresden, Institut für Mathematische Stochastik, Helmholtzstraße 10, 01069 Dresden, Germany
*Postal address: Technische Universität Dresden, Institut für Mathematische Stochastik, Helmholtzstraße 10, 01069 Dresden, Germany

Abstract

The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether it is drifting to $+\infty$ , $-\infty$ , or oscillating. Whenever the Lévy process drifts to $+\infty$ , we prove that the $\kappa$ th moment of the first passage time (conditioned to be finite) exists if and only if the $(\kappa+1)$ th moment of the Lévy jump measure exists. This generalizes a result shown earlier by Delbaen for Cramér–Lundberg risk processes. Whenever the Lévy process drifts to $-\infty$ , we prove that all moments of the first passage time exist, while for an oscillating Lévy process we derive conditions for non-existence of the moments, and in particular we show that no integer moments exist.

Type
Original Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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