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Parisian ruin for a refracted L\'evy process

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  • Mohamed Amine Lkabous
  • Irmina Czarna
  • Jean-Franc{c}ois Renaud

Abstract

In this paper, we investigate Parisian ruin for a L\'evy surplus process with an adaptive premium rate, namely a refracted L\'evy process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Our main contribution is a generalization of the result in Loeffen et al. (2013) for the probability of Parisian ruin of a standard L\'evy insurance risk process. Despite the more general setup considered here, our main result is as compact and has a similar structure. Examples are provided.

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  • Mohamed Amine Lkabous & Irmina Czarna & Jean-Franc{c}ois Renaud, 2016. "Parisian ruin for a refracted L\'evy process," Papers 1603.09324, arXiv.org, revised Mar 2017.
    • Handle: RePEc:arx:papers:1603.09324
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    References listed on IDEAS

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    1. David Landriault & Jean-François Renaud & Xiaowen Zhou, 2014. "An Insurance Risk Model with Parisian Implementation Delays," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 583-607, September.
    2. Loeffen, Ronnie L. & Renaud, Jean-François & Zhou, Xiaowen, 2014. "Occupation times of intervals until first passage times for spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1408-1435.
    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.
    4. Wong, Jeff T.Y. & Cheung, Eric C.K., 2015. "On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 280-290.
    5. 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.
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    Cited by:

    1. Mohamed Amine Lkabous, 2019. "A note on Parisian ruin under a hybrid observation scheme," Papers 1907.09993, arXiv.org.
    2. Irmina Czarna & Adam Kaszubowski, 2020. "Optimality of Impulse Control Problem in Refracted Lévy Model with Parisian Ruin and Transaction Costs," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 982-1007, June.
    3. David Landriault & Bin Li & Mohamed Amine Lkabous, 2019. "On occupation times in the red of L\'evy risk models," Papers 1903.03721, arXiv.org, revised Jul 2019.

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