Skip to main content Accessibility help
×
Home

Large deviations of heavy-tailed random sums with applications in insurance and finance

  • C. Klüppelberg (a1) and T. Mikosch (a2)

Abstract

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

Copyright

Corresponding author

Postal address: Department of Mathematics, Johannes Gutenberg University Mainz, D-55099 Mainz, Germany.
∗∗ Postal address: Department of Mathematics, University of Groningen, P.O. Box 800, NL-9700 Groningen, The Netherlands.

References

Hide All
Aase, K. (1994) An equilibrium model of catastrophic insurance futures contracts. Preprint.
Albrecht, P., König, A. and Schradin, H. R. (1994) Katastrophenversicherungs-Termingeschäfte: Grundlagen und Anwendungen im Risikomanagement von Versicherungsunternehmen. Mannheimer Manuskripte zur Versicherungsbetriebslehre, Finanzmanagement und Risikotheorie. No. 72. Universität Mannheim.
Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer, Berlin.
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Cambridge University Press, Cambridge.
Chicago Board Of Trade (Cbot) (1992) Homeowners futures and hedging insurance price risk.
Cistyakov, V. P. (1964) A theorem on the sums of independent positive random variables and its applications to branching random processes. Theory Prob. Appl. 9, 640648.
Cline, D. B. and Hsing, T. (1991) Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint. Texas A & M University.
Cummins, J. D. and Geman, H. (1994) An Asian option approach to the valuation of insurance futures contracts. Rev. Futures Markets 13, 517557.
Embrechts, P., Goldie, C. M. and Veraverbere, N. (1979) Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335347.
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
Embrechts, P. and Veraverbere, N. (1982) Estimates for the probability of ruin with special emphasis on the possibility of large claims. Ins. Math. Econ. 1, 5572.
Gut, A. (1988) Stopped Random Walk. Limit Theorems and Applications. Springer, New York.
Heyde, C. C. (1967a) A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitsth. 7, 303308.
Heyde, C. C. (1967b) On large deviation problems for sums of random variables which are not attracted to the normal law. Ann. Math. Statist. 38, 15751578.
Heyde, C. C. (1968) On large deviation probabilities in the case of attraction to a nonnormal stable law. Sankhya 30, 253258.
Meister, S. (1995) Contributions to the mathematics of catastrophe insurance futures. Diplomarbeit. ?TH, Zürich.
Nagaev, A. V. (1969a) Integral limit theorems for large deviations when Cramér's condition is not fulfilled I, II. Theory Prob. Appl. 14, 5164, 193-208.
Nagaev, A. V. (1969b) Limit theorems for large deviations when Cramér's conditions are violated. (In Russian.) Fiz-Mat. Nauk. 7, 1722.
Nagaev, S. V. (1973) Large deviations for sums of independent random variables. In Trans. Sixth Prague Conf. on Information Theory, Random Processes and Statistical Decision Functions. Academia, Prague. pp. 657674.
Nagaev, S. V. (1979) Large deviations of sums of independent random variables. Ann. Prob. 7, 745789.
Pakes, A. G. (1975) On the tails of waiting-time distributions. J. Appl. Prob. 12, 555564.
Petrov, V. V. (1975) Sums of Independent Random Variables. Springer, Berlin.
Pinelis, I. F. (1985) On the asymptotic equivalence of probabilities of large deviations for sums and maxima of independent random variables. (In Russian.) In Limit Theorems in Probability Theory. (Trudy Inst. Math. 5.) Nauka, Novosibirsk.

Keywords

MSC classification

Large deviations of heavy-tailed random sums with applications in insurance and finance

  • C. Klüppelberg (a1) and T. Mikosch (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed