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Ruin probability with Parisian delay for a spectrally negative L\'evy risk process

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  • Irmina Czarna
  • Zbigniew Palmowski

Abstract

In this paper we analyze so-called Parisian ruin probability that happens when surplus process stays below zero longer than fixed amount of time $\zeta>0$. We focus on general spectrally negative L\'{e}vy insurance risk process. For this class of processes we identify expression for ruin probability in terms of some other quantities that could be possibly calculated explicitly in many models. We find its Cram\'{e}r-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.

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  • Irmina Czarna & Zbigniew Palmowski, 2010. "Ruin probability with Parisian delay for a spectrally negative L\'evy risk process," Papers 1003.4299, arXiv.org, revised Apr 2010.
    • Handle: RePEc:arx:papers:1003.4299
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    References listed on IDEAS

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    1. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
    2. Bertoin, J. & Doney, R. A., 1994. "Cramer's estimate for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 21(5), pages 363-365, December.
    3. Wang, Guojing, 2001. "A decomposition of the ruin probability for the risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 49-59, February.
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    Cited by:

    1. Irmina Czarna & Zbigniew Palmowski, 2010. "Dividend problem with Parisian delay for a spectrally negative L\'evy risk process," Papers 1004.3310, arXiv.org, revised Oct 2011.
    2. Ronnie Loeffen & Irmina Czarna & Zbigniew Palmowski, 2011. "Parisian ruin probability for spectrally negative L\'{e}vy processes," Papers 1102.4055, arXiv.org, revised Mar 2013.
    3. Landriault, David & Renaud, Jean-François & Zhou, Xiaowen, 2011. "Occupation times of spectrally negative Lévy processes with applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2629-2641, November.

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