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Moment and MGF convergence of overshoots and undershoots for Lévy insurance risk processes

Part of: Mathematical economics Limit theorems Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Hyun Suk Park  and
Ross Maller
Hyun Suk Park*
Affiliation:
Australian National University and Pohang University of Science and Technology
Ross Maller*
Affiliation:
Australian National University
*
∗ Postal address: Pohang Mathematics Institute, POSTECH, Pohang 790-784, South Korea. Email address: [email protected]
∗∗ Postal address: School of Finance and Applied Statistics and Centre for Mathematics and its Applications, Australian National University, Canberra 0200, Australia. Email address: [email protected]
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Abstract

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This paper is concerned with the finiteness and large-time behaviour of moments of the overshoot and undershoot of a high level, and of their moment generating functions (MGFs), for a Lévy process which drifts to -∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process. Results of Klüppelberg, Kyprianou, and Maller (2004) and Doney and Kyprianou (2006) for asymptotic overshoot and undershoot distributions in the class of Lévy processes with convolution equivalent canonical measures are shown to have moment and MGF convergence extensions.

Keywords

Insurance risk processovershootundershootLévy processladder height processsubexponential and convolution equivalent distributions

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60K05: Renewal theory
Secondary: 91B30: Risk theory, insurance 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 716 - 733
Copyright
Copyright © Applied Probability Trust 2008 

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