Hostname: page-component-857557d7f7-nhjpk Total loading time: 0 Render date: 2025-11-20T10:22:35.971Z Has data issue: false hasContentIssue false
  • English
  • Français

Ruin Probability with Parisian Delay for a Spectrally Negative Lévy Risk Process

Part of: Stochastic processes Stochastic systems and control Markov processes

Published online by Cambridge University Press:  14 July 2016

Irmina Czarna  and
Zbigniew Palmowski
Irmina Czarna*
Affiliation:
University of Wrocław
Zbigniew Palmowski*
Affiliation:
University of Wrocław
*
∗ Postal address: Department of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
∗ Postal address: Department of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples.

Keywords

Lévy processruin probabilityasymptoticsParisian ruinrisk process

MSC classification

Secondary: 60J99: None of the above, but in this section 93E20: Optimal stochastic control 60G51: Processes with independent increments; L'evy processes

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 984 - 1002
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Albrecher, H., Kortschak, D. and Zhou, X. (2010). Pricing of Parisian options for a Jump-diffusion model with two-sided Jumps. Submitted.Google Scholar
[2] Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ.Google Scholar
[3] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[4] Bertoin, J. and Doney, R. A. (1994). Cramér's estimate for Lévy processes. Statist. Prob. Lett. 21, 363–365.Google Scholar
[5] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
[6] Borovkov, K. and Burq, Z. (2001). Kendall's identity for the first crossing time revisited. Electron. Commun. Prob. 6, 91–94.Google Scholar
[7] Chesney, M., Jeanblanc-Picqué, M. and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. Appl. Prob. 29, 165–184.Google Scholar
[8] Dassios, A. and Wu, S. (2009). Parisian ruin with exponential claims. Submitted. Available at http://stats. lse.ac.uk/angelos/.Google Scholar
[9] Dassios, A. and Wu, S. (2009). Ruin probabilities of the Parisian type for small claims. Submitted. Available at http://stats.lse.ac.uk/angelos/.Google Scholar
[10] Dassios, A. and Wu, S. (2010). Perturbed Brownian motion and its application to Parisian option pricing. Finance Stoch. 14, 473–494.Google Scholar
[11] Dufresne, F. and Gerber, H. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10, 51–59.Google Scholar
[12] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1954). Tables of Integral Transforms, Vol. I. McGraw-Hill, New York.Google Scholar
[13] Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuarial J. 1998, 59–74.Google Scholar
[14] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities and decompositions for general perturbed risk process. Ann. Appl. Prob. 14, 1378–1397.Google Scholar
[15] Klüppelberg, C. and Kyprianou, A. E. (2006). On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Prob. 43, 594–598.Google Scholar
[16] Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.Google Scholar
[17] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[18] Kyprianou, A. E. and Palmowski, Z. (2005). A martingale review of some fluctuation theory for spectrally negative Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 16–29.Google Scholar
[19] Landriault, D., Renaud, J. F. and Zhou, X. (2011). Insurance risk models with Parisian implementation delays. Submitted. Available at http://www.math.uqam.ca/∼renaudjf/recherche.html.Google Scholar
[20] Landriault, D., Renaud, J. F. and Zhou, X. (2011). Occupation times of spectrally negative Lévy processes with applications. Submitted. Available at http://www.math.uqam.ca/∼renaudjf/recherche.html.Google Scholar
[21] Loeffen, R., Czarna, I. and Palmowski, Z. (2011). Parisian ruin probability for spectrally negative Lévy processes. Submitted. Available at http://arxiv.org/abs/1102.4055v1.Google Scholar
[22] Wang, G. (2001). A decomposition of the ruin probability for the risk process perturbed by diffusion. Insurance Math. Econom. 28, 49–59.Google Scholar