×

zbMATH — the first resource for mathematics

Parisian ruin probability for spectrally negative Lévy processes. (English) Zbl 1267.60054
Summary: We give, for a spectrally negative Lévy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period \(r\). The formula involves only the scale function of the spectrally negative Lévy process and the distribution of the process at time \(r\).

MSC:
60G51 Processes with independent increments; Lévy processes
91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121 . Cambridge: Cambridge Univ. Press. · Zbl 0861.60003
[2] Chazal, M., Kyprianou, A.E. and Patie, P. (2010). A transformation for Lévy processes with one-sided jumps and applications. Preprint. Available at . 1010.3819v1 · arXiv.org
[3] Chesney, M., Jeanblanc-Picqué, M. and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. in Appl. Probab. 29 165-184. · Zbl 0882.60042 · doi:10.2307/1427865
[4] Czarna, I. and Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk process. J. Appl. Probab. 48 984-1002. · Zbl 1232.60036 · doi:10.1239/jap/1324046014
[5] Dassios, A. and Wu, S. Semi-Markov model for excursions and occupation time of Markov processes. Working paper, LSE, London. Available at . · stats.lse.ac.uk
[6] Dassios, A. and Wu, S. Parisian ruin with exponential claims. Working paper, LSE, London. Available at . · stats.lse.ac.uk
[7] Kyprianou, A.E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext . Berlin: Springer. · Zbl 1104.60001
[8] Landriault, D. Renaud, J.-F. and Zhou, X. (2010). Insurance risk models with Parisian implementation delays. · Zbl 1319.60098
[9] Lebedev, N.N. (1965). Special Functions and Their Applications , revised English ed. Englewood Cliffs, NJ: Prentice-Hall Inc. Translated and edited by Richard A. Silverman. · Zbl 0131.07002
[10] Loeffen, R.L. and Renaud, J.F. (2010). De Finetti’s optimal dividends problem with an affine penalty function at ruin. Insurance Math. Econom. 46 98-108. · Zbl 1231.91212 · doi:10.1016/j.insmatheco.2009.09.006
[11] Schneider, W.R. (1986). Stable distributions: Fox functions representation and generalization. In Stochastic Processes in Classical and Quantum Systems ( Ascona , 1985). Lecture Notes in Physics 262 497-511. Berlin: Springer. · doi:10.1007/3540171665_92
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.