Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T08:22:21.766Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential Behavior in the Presence of Dependence in Risk Theory

Part of: Stochastic processes Special processes Applications

Published online by Cambridge University Press:  14 July 2016

Hansjörg Albrecher  and
Jef L. Teugels
Hansjörg Albrecher*
Affiliation:
Austrian Academy of Sciences, Linz, and Graz University of Technology
Jef L. Teugels*
Affiliation:
Katholieke Universiteit Leuven and EURANDOM
*
∗ Postal address: Department of Mathematics, Graz University of Technology, Steyrergasse 30, Graz, 8010, Austria. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, Katholieke Universiteit Leuven, de Croylaan 54, Heverlee, 3001, Belgium.
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.

We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.

Keywords

Dependencerisk modelcopularenewal theoryWiener-Hopf theory

MSC classification

Primary: 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 60G50: Sums of independent random variables; random walks 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 257 - 273
Copyright
© Applied Probability Trust 2006 

References

Asmussen, S. (1982). Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue. Adv. Appl. Prob. 14, 143–170.Google Scholar
Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.Google Scholar
Baltrūnas, A. (2001). Some asymptotic results for transient random walks with applications to insurance risk. J. Appl. Prob. 38, 108–121.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and its Applications, Vol. 2. John Wiley, New York.Google Scholar
Genest, C. and Rivest, L. (1993). Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88, 1034–1043.Google Scholar
Hürlimann, W. (2000). A Spearman multivariate distribution with fixed margins – theory and applications. Preprint.Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.Google Scholar
Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions, Vol. 1, Models and Applications. John Wiley, New York.Google Scholar
Nelsen, R. (1999). An Introduction to Copulas. Springer, Berlin.Google Scholar
Prabhu, N. U. (1965). Stochastic Processes. Basic Theory and its Applications. Macmillan, New York.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. L. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
Veraverbeke, N. and Teugels, J. L. (1975). The exponential rate of convergence of the distribution of the maximum of a random walk. J. Appl. Prob. 12, 279–288.Google Scholar
Veraverbeke, N. and Teugels, J. L. (1976). The exponential rate of convergence of the distribution of the maximum of a random walk. II. J. Appl. Prob. 13, 733–740.Google Scholar
Whittaker, E. and Watson, G. (1963). A Course of Modern Analysis, 4th edn. Cambridge University Press.Google Scholar
Widder, D. (1942). The Laplace Transform. Princeton University Press.Google Scholar