Science in China Series A: Mathematics

, Volume 45, Issue 5, pp 632–639 | Cite as

An asymptotic relationship for ruin probabilities under heavy-tailed claims

Article
  • 40 Downloads

Abstract

The famous Embrechts-Goldie-Veraverbeke formula shows that, in the classical Cramér-Lundberg risk model, the ruin probabilities satisfy \(R(x, \infty ) \sim \rho ^{ - 1} \bar F_e (x)\) if the claim sizes are heavy-tailed, where Fe denotes the equilibrium distribution of the common d.f. F of the i.i.d. claims, ? is the safety loading coefficient of the model and the limit process is for x → ∞. In this paper we obtain a related local asymptotic relationship for the ruin probabilities. In doing this we establish two lemmas regarding the n-fold convolution of subexponential equilibrium distributions, which are of significance on their own right.

Keywords

Cramér-Lundberg model geometric sums heavy-tailed distribution ladder height ruin probabilities 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Embrechts, P., Klüppelberg, C., Mikosch, T., Modelling Extremal Events for Insurance and Finance, Berlin: Springer, 1997.MATHGoogle Scholar
  2. 2.
    Ross, S. M., Stochastic Processes, New York: Wiley, 1983.MATHGoogle Scholar
  3. 3.
    Rolski, T., Schmidli, H., Schmidt, V. et al., Stochastic Processes for Insurance and Finance, New York: Wiley, 1999.MATHGoogle Scholar
  4. 4.
    Asmussen, S., Ruin Probabilities, Singapore: World Scientific, 2000.Google Scholar
  5. 5.
    Klüppelberg, C., Subexponential distributions and integrated tails, J. Appl. Prob., 1988, 25: 132.MATHCrossRefGoogle Scholar
  6. 6.
    Embrechts, P., Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: Mathematics and Economics, 1982, 1: 55.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Tang, Q. H., Su, C., Ruin probabilities for large claims in delayed renewal risk model, Southeast Asian Bull. Math., 2001, 25(4).Google Scholar
  8. 8.
    Asmussen, S., Subexpontential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities, the Ann. of Appl. Prob., 1998, 8: 354.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Asmussen, S., Applied Probability and Queues, New York: John Wiley & Sons, 1987.MATHGoogle Scholar
  10. 10.
    Tang, Q. H., Extremal values of risk processes for insurance and finance: with special emphasis on the possibility of large claims, Doctoral thesis of University of Science and Technology of China, 2001.Google Scholar
  11. 11.
    Kalashnikov, V. V., Konstantinides, D., A simple proof of a result of Asmussen, Working paper 160, Lab. of Actuarial Math., University of Copenhagen, 1999.Google Scholar
  12. 12.
    Kalashnikov, V. V., Konstantinides, D., Ruin under interest force and subexponential claims: a simple treatment, Insurance: Mathematics and Economics, 2000, 27: 145.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cistyakov, V. P., A theorem on the sums of independent positive random variables and its applications to branching random processes, Theory Prob. Appl., 1964, 9: 640.CrossRefGoogle Scholar
  14. 14.
    Athreya, K. B., Ney, P. E., Branching Processes, Berlin: Springer, 1972.MATHGoogle Scholar
  15. 15.
    Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed., New York: Wiley, 1971.MATHGoogle Scholar
  16. 16.
    Veraverbeke, N., Asymptotic behavior of Wiener-Hopf factors of a random walk, Stoch. Proc. Appl., 1977, 5: 27.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kalashnikov, V. V., Two-sided bounds of ruin probabilities, Scand. Act. J., 1996, 1: 1.MathSciNetGoogle Scholar
  18. 18.
    Kalashnikov, V. V., Geometric Sums: Bounds for Rare Events with Applications, Dordrecht: Kluwer Acad. Publ., 1997.MATHGoogle Scholar

Copyright information

© Science in China Press 2002

Authors and Affiliations

  1. 1.Department of Statistics and FinanceUniversity of Science and Technology of ChinaHefeiChina

Personalised recommendations