Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T06:22:59.085Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes Limit theorems Special processes Mathematical economics

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: hspark@postech.ac.kr
∗∗ Postal address: School of Finance and Applied Statistics and Centre for Mathematics and its Applications, Australian National University, Canberra 0200, Australia. Email address: ross.maller@anu.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Doney, R. A. and Kyprianou, A. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16, 91–106.CrossRefGoogle Scholar
Doney, R. A. and Maller, R. A. (2004). Moments of passage times for Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 40, 279–297.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Process. Appl. 13, 263–278.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.CrossRefGoogle Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.CrossRefGoogle Scholar
Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82, 259–269.CrossRefGoogle Scholar
Klüppelberg, C., Kyprianou, A. and Maller, R. (2004). Ruin probability and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.CrossRefGoogle Scholar
Kyprianou, A. (2005). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Vigon, V. (2002). Votre Lévy rampe-t-il? J. London Math. Soc. 65, 243–256.CrossRefGoogle Scholar