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An Extension of Panjer's Recursion

  • Klaus Th. Hess (a1), Anett Liewald (a1) and Klaus D. Schmidt (a1)
Abstract
Abstract

Sundt and Jewell have shown that a nondegenerate claim number distribution Q = {qn}nϵN0 satisfies the recursion

for all n≥0 if and only if Q is a binomial, Poisson or negativebinomial distribution. This recursion is of interest since it yields a recursion for the aggregate claims distribution in the collective model of risk theory when the claim size distribution is integer-valued as well. A similar characterization of claim number distributions satisfying the above recursion for all n ≥ 1 has been obtained by Willmot. In the present paper we extend these results and the subsequent recursion for the aggregate claims distribution to the case where the recursion holds for all nk with arbitrary k. Our results are of interest in catastrophe excess-of-loss reinsurance.

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Copyright
Corresponding author
Lehrstuhl für Versicherungsmathemetik, Technische Universität Dresden, D-01062 Dresden E-mail:schmidt@math.tu-dresden.de
References
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Hofmann M. (1955) Über zusammengesetzte Poisson-Prozesse und ihre Anwendungen in der Unfallversicherung. Mitt. Verein. Schweiz. Versicherungsmathematiker 55, 499575.
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Klugman S.A., Panjer H.H. and Willmot G.E. (1998) Loss Models. Wiley, New York and Chichester.
Panjer H.H. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bull. 12, 2226.
Schmidt K.D. (1996) Lectures on Risk Theory. Teubner, Stuttgart.
Schmidt K.D. (2001) Versicherungsmathematik. Springer, Berlin - Heidelberg - New York.
Sundt B. (1992) On some extensions of Panjer's class of counting distributions. ASTIN Bull. 22, 6180.
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Willmot G.E. (1988) Sundt and Jewell's family of discrete distributions. ASTIN Bull. 18, 1729.
Willmot G.E. and Lin X.S. (2001) Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, Berlin - Heidelberg - New York.
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ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
  • URL: /core/journals/astin-bulletin-journal-of-the-iaa
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