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Poissonian occupation times of spectrally negative L\'evy processes with applications

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  • Mohamed Amine Lkabous

Abstract

In this paper, we introduce the concept of \emph{Poissonian occupation times} below level $0$ of spectrally negative L\'evy processes. In this case, occupation time is accumulated only when the process is observed to be negative at arrival epochs of an independent Poisson process. Our results extend some well known continuously observed quantities involving occupation times of spectrally negative L\'evy processes. As an application, we establish a link between Poissonian occupation times and insurance risk models with Parisian implementation delays.

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  • Mohamed Amine Lkabous, 2019. "Poissonian occupation times of spectrally negative L\'evy processes with applications," Papers 1907.09990, arXiv.org.
    • Handle: RePEc:arx:papers:1907.09990
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    References listed on IDEAS

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    1. 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.
    2. Mohamed Amine Lkabous, 2019. "A note on Parisian ruin under a hybrid observation scheme," Papers 1907.09993, arXiv.org.
    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. Li, Yingqiu & Zhou, Xiaowen, 2014. "On pre-exit joint occupation times for spectrally negative Lévy processes," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 48-55.
    5. Lkabous, Mohamed Amine, 2019. "A note on Parisian ruin under a hybrid observation scheme," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 147-157.
    6. Dickson,David C. M., 2016. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9781107154605.
    7. Albrecher, Hansjörg & Ivanovs, Jevgenijs, 2017. "Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 643-656.
    8. Li, Yingqiu & Zhou, Xiaowen & Zhu, Na, 2015. "Two-sided discounted potential measures for spectrally negative Lévy processes," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 67-76.
    9. 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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