Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T00:19:47.365Z Has data issue: false hasContentIssue false
  • English
  • Français

On extreme ruinous behaviour of Lévy insurance risk processes

Part of: Special processes Markov processes Stochastic processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

C. Klüppelberg  and
A. E. Kyprianou
C. Klüppelberg*
Affiliation:
Munich University of Technology
A. E. Kyprianou*
Affiliation:
Heriot-Watt University
*
∗Postal address: Center for Mathematical Sciences, Munich University of Technology, Boltzmannstrasse 3, D-85747 Garching, Germany. Email address: [email protected]
∗∗Postal address: School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: [email protected]
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.

In this short note we show how new fluctuation identities and their associated asymptotics, given in Vigon (2002), Klüppelberg et al. (2004) and Doney and Kyprianou (2006), provide the basis for establishing, in an elementary way, asymptotic overshoot and undershoot distribitions for a general class of Lévy insurance risk processes. The results bring the earlier conclusions of Asmussen and Klüppelberg (1996) for the Cramér-Lundberg process into greater generality.

Keywords

Lévy processinsurance risk processruinextreme value theory

MSC classification

Primary: 60K05: Renewal theory 60K15: Markov renewal processes, semi-Markov processes 91B30: Risk theory, insurance
Secondary: 60G70: Extreme value theory; extremal processes 60J55: Local time and additive functionals
Type
Short Communications
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 594 - 598
Copyright
© Applied Probability Trust 2006 

References

Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Process. Appl. 64, 103–125.CrossRefGoogle Scholar
Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16, 91–106.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions. In A Practical Guide to Heavy Tails, eds Adler, R., Feldman, R. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 435–459.Google Scholar
Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004a). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14, 1378–1397.Google Scholar
Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004b). Ruin probabilities for competing claim processes. J. Appl. Prob. 41, 679–690.CrossRefGoogle Scholar
Klüppelberg, C., Kyprianou, A. E. and Maller, R. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.Google Scholar
Vigon, V. (2002). Votre Lévy rampe-t-il? J. London Math. Soc. 65, 243–256.CrossRefGoogle Scholar