The Annals of Probability

Ruin probability with claims modeled by a stationary ergodic stable process

Thomas Mikosch and Gennady Samorodnitsky

Full-text: Open access

Abstract

For a random walk with negative drift we study the exceedance probability (ruin probability) of a high threshold. The steps of this walk (claim sizes) constitute a stationary ergodic stable process. We study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes. Our findings show that the order of magnitude of the ruin probability varies significantly from one model to another. In particular, ruin becomes much more likely when the claim sizes exhibit long-range dependence. The proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a Poisson process.

Article information

Source
Ann. Probab., Volume 28, Number 4 (2000), 1814-1851.

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1019160509

Digital Object Identifier
doi:10.1214/aop/1019160509

Mathematical Reviews number (MathSciNet)
MR1813844

Zentralblatt MATH identifier
1044.60028

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60G10: Stationary processes 60K30

Keywords
Stable process stationary process ruin probability heavy tails supremum negative drift risk

Citation

Mikosch, Thomas; Samorodnitsky, Gennady. Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Probab. 28 (2000), no. 4, 1814--1851. doi:10.1214/aop/1019160509. https://projecteuclid.org/euclid.aop/1019160509


Export citation

References

  • [1] Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probabilities for non-standard risk and queueing processes with subexponential jumps. Adv. Appl. Probab. 31 422-447.
  • [2] Baccelli, F. and Br´emaud, P. (1994). Elements of Queueing Theory. Palm-Martingale Calculus and Stochastic Recurrences. Springer, Berlin.
  • [3] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [4] Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation. Cambridge Univ. Press.
  • [5] Braverman, M., Mikosch, T. and G. Samorodnitsky (2000). The tail behaviour of subadditive functionals acting on L´evy processes. Preprint available as levy.ps from http://www.math.rug.nl/ mikosch/Connie.
  • [6] Crovella, M. and Bestavros, A. (1995). Explaining world wide web traffic self-similarity. Preprint available as TR-95-015 from crovella,best cs.bu.edu.
  • [7] Embrechts, P., Kl ¨uppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  • [8] Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Mathematics and Economics 1 55-72.
  • [9] Goldie, C. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126-166.
  • [10] Gross, A. (1994). Some mixing conditions for stationary symmetric stable stochastic processes. Stochastic Process. Appl. 51 277-295.
  • [11] Kallenberg, O. (1983). RandomMeasures, 3rd. ed. Akademie, Berlin.
  • [12] Kasahara, Y., Maejima, M. and Vervaat, W. (1988). Log-fractional stable processes. Stochastic Process. Appl. 30 329-339.
  • [13] Maejima, M. (1983). A self-similar process with nowhere bounded sample paths.Wahrsch. Verw. Gebiete 65 115-119.
  • [14] Maruyama, G. (1970). Infinitely divisible processes. Theory Probab. Appl. 15 1-22.
  • [15] Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 10 1025-1064.
  • [16] Mittnik, S. and Rachev, S. (1993). Modeling asset returns with alternative stable distributions (with discussion). Econom. Rev. 12 261-389.
  • [17] Rajput, B. and Rosi ´nski, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 451-488.
  • [18] Resnick, S. (1997). Discussion of the Danish data on large fire insurance losses. ASTIN Bull. 27 139-151.
  • [19] Resnick, S., Samorodnitsky, G. and Xue, F. (2000). Growth rates of sample covariances of stationary symmetric -stable processes associated with null recurrent Markov chains. Stochastic Process. Appl. 85 321-339.
  • [20] Rosi ´nski, J. (1986). On stochastic integral representation of stable processes with sample paths in Banach spaces. J. Multivariate Anal. 20 277-302.
  • [21] Rosi ´nski, J. (1995). On the structure of stationary stable processes. Ann. Probab. 23 1163- 1187.
  • [22] Rosi ´nski, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 996-1014.
  • [23] Rosi ´nski, J. and Samorodnitsky, G. (1996). Classes of mixing stable processes. Bernoulli 2 3655-378.
  • [24] Samorodnitsky, G. and Taqqu, M. (1990). 1/ -self-similar processes with stationary increments. J. Multivariate Anal. 35 308-313.
  • [25] Samorodnitsky, G. and Taqqu, M. (1994). Stable Non-Gaussian RandomProcesses. Chapman and Hall, New York.
  • [26] Surgailis, D., Rosi ´nski, J., Mandrekar, V. and Cambanis, S. (1993). Stable mixed moving averages. Probab. Theory Related Fields 97 543-558.
  • [27] Taqqu, M. (1988). Self-similar processes. In Encyclopedia of Statistical Sciences 8 (S. Kotz and N. Johnson, eds.) 352-257. Wiley, New York.
  • [28] Taqqu, M. and Wolpert, R. (1983). Infinite variance self-similar processes subordinate to a Poisson measure.Wahrsch. Verw. Gebiete 62 53-72.
  • [29] Teicher, H. (1979). Rapidly growing random walks and an associated stopping time. Ann. Probab. 7 1078-1081.
  • [30] Vervaat, W. (1985). Sample paths of self-similar processes with stationary increments. Ann. Probab. 13 1-27.
  • [31] Willinger, W., Taqqu, M., Leland, M. and Wilson, D. (1995). Self-similarity through high variability: statistical analysis of ethernet LAN traffic at the source level. Comput. Comm. Rev. 25 100-113.