Tail probabilities of subadditive functionals of Lévy processes. (English) Zbl 1035.60045

Braverman, Michael; Mikosch, Thomas; Samorodnitsky, Gennady

Ann. Appl. Probab. 12, No. 1, 69-100 (2002).

Let \(X = X (t)\) denote a Lévy process without Gaussian component for \(t \geq 0\) on the real line and let \(\varphi \) denote a subadditive functional on the sample paths. The paper is concerned with the exceedance probability \(P (\varphi(X-\mu) > u)\), where \(\mu = \mu (t)\) is a deterministic curve. Under various conditions these probabilities behave like a known regularly varying function \(u \mapsto \varphi (u)\) if \(u\) tends to infinity. A proper choice of the functional \(\varphi \) yields concrete application. The authors establish the asymptotic decay of the probability of the following events: the time the process spends above zero, the last hitting time of zero, the supremum of the integral process greater than \(u\). The results include the decay of ruin probabilities.

Reviewer: Arnold Janssen (Düsseldorf)

MSC:

60G51	Processes with independent increments; Lévy processes
60E07	Infinitely divisible distributions; stable distributions

Keywords:

ruin probability; heavy tails; supremum; negtive drift; risk

References:

[1]	BERTOIN, J. (1996). Lévy Processes. Cambridge Univ. Press. · Zbl 0861.60003
[2]	BINGHAM N. H., GOLDIE, C. M. and TEUGELS, J. L. (1987). Regular Variation. Cambridge Univ. Press. · Zbl 0617.26001
[3]	BRAVERMAN, M., MIKOSCH, T. and SAMORODNITSKY, G. (2000). Tail probabilities of subadditive functionals acting on Lévy processes. Technical report, Laboratory of Actuarial Mathematics, Univ. Copenhagen. Available as levy.ps.gz under www.math.ku.dk/ mikosch/Preprint/Levy. URL:
[4]	EMBRECHTS, P., GOLDIE, C. and VERAVERBEKE, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 49 335-347. · Zbl 0397.60024 · doi:10.1007/BF00535504
[5]	EMBRECHTS, P., KLÜPPELBERG, C. and MIKOSCH, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin. · Zbl 0873.62116
[6]	EMBRECHTS, P. and VERAVERBEKE, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55-72. · Zbl 0518.62083 · doi:10.1016/0167-6687(82)90021-X
[7]	HÜSLER, J. and PITERBARG, V. (1999). Extremes of a certain class of Gaussian processes. Stochastic Process. Appl. 83 257-271. · Zbl 0997.60057 · doi:10.1016/S0304-4149(99)00041-1
[8]	MARUYAMA, G. (1970). Infinitely divisible processes. Theory Probab. Appl. 15 1-22. · Zbl 0268.60036
[9]	MIKOSCH, T. and SAMORODNITSKY, G. (2000). Ruin probabilities for a random walk with stable stationary ergodic increments. Ann. Probab. 28 1814-1851. · Zbl 1044.60028 · doi:10.1214/aop/1019160509
[10]	PETROV, V. (1995). Limit Theorems of Probability Theory. Oxford Univ. Press. · Zbl 0826.60001
[11]	PRABHU, N. U. (1998). Stochastic Storage Processes. Queues, Insurance Risk, Dams, and Data Communication, 2nd ed. Springer, New York.
[12]	PROKHOROV, Y. (1959). An extremal problem in probability theory. Theory Probab. Appl. 4 201- 204. · Zbl 0093.15102 · doi:10.1137/1104017
[13]	RESNICK, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York. · Zbl 0633.60001
[14]	ROSI ŃSKI, J. and SAMORODNITSKY, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 996-1014. · Zbl 0776.60049 · doi:10.1214/aop/1176989279
[15]	SATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press. · Zbl 0973.60001
[16]	STOUT, W. F. (1974). Almost Sure Convergence. Academic Press, New York. · Zbl 0321.60022
[17]	ITHACA, NEW YORK 14853 E-MAIL: [e-mail]