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Spectrally negative Lévy processes with Parisian reflection below and classical reflection above

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  • Avram, Florin
  • Pérez, José-Luis
  • Yamazaki, Kazutoshi

Abstract

We consider a company that receives capital injections so as to avoid ruin. Differently from the classical bail-out settings, where the underlying process is restricted to stay at or above zero, we study the case bail-out can only be made at independent Poisson observation times. Namely, we study a version of the reflected process that is pushed up to zero only on Poisson arrival times at which the process is below zero. We also study the case with additional classical reflection above so as to model a company that pays dividends according to a barrier strategy. Focusing on the spectrally negative Lévy case, we compute, using the scale function, various fluctuation identities, including capital injections and dividends.

Suggested Citation

  • Avram, Florin & Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "Spectrally negative Lévy processes with Parisian reflection below and classical reflection above," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 255-290.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:1:p:255-290
    DOI: 10.1016/j.spa.2017.04.013
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    References listed on IDEAS

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    1. Pérez, José-Luis & Yamazaki, Kazutoshi, 2017. "Refraction–Reflection Strategies In The Dual Model," ASTIN Bulletin, Cambridge University Press, vol. 47(1), pages 199-238, January.
    2. Avanzi, Benjamin & Cheung, Eric C.K. & Wong, Bernard & Woo, Jae-Kyung, 2013. "On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 98-113.
    3. Dickson,David C. M., 2016. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9781107154605.
    4. Florin Avram & Zbigniew Palmowski & Martijn R. Pistorius, 2007. "On the optimal dividend problem for a spectrally negative L\'{e}vy process," Papers math/0702893, arXiv.org.
    5. 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.
    6. Hansjörg Albrecher & Jevgenijs Ivanovs, 2013. "A Risk Model with an Observer in a Markov Environment," Risks, MDPI, vol. 1(3), pages 1-14, November.
    7. 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.
    8. Albrecher, Hansjörg & Cheung, Eric C.K. & Thonhauser, Stefan, 2011. "Randomized Observation Periods for the Compound Poisson Risk Model: Dividends," ASTIN Bulletin, Cambridge University Press, vol. 41(2), pages 645-672, November.
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    Cited by:

    1. Noba, Kei & Yano, Kouji, 2019. "Generalized refracted Lévy process and its application to exit problem," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1697-1725.
    2. Wenyuan Wang & Yuebao Wang & Ping Chen & Xueyuan Wu, 2022. "Dividend and Capital Injection Optimization with Transaction Cost for Lévy Risk Processes," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 924-965, September.
    3. Dong, Hua & Zhou, Xiaowen, 2019. "On a spectrally negative Lévy risk process with periodic dividends and capital injections," Statistics & Probability Letters, Elsevier, vol. 155(C), pages 1-1.
    4. Landriault, David & Li, Bin & Lkabous, Mohamed Amine, 2021. "On the analysis of deep drawdowns for the Lévy insurance risk model," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 147-155.
    5. Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "American options under periodic exercise opportunities," Statistics & Probability Letters, Elsevier, vol. 135(C), pages 92-101.
    6. Zbigniew Palmowski & Jos'e Luis P'erez & Budhi Arta Surya & Kazutoshi Yamazaki, 2019. "The Leland-Toft optimal capital structure model under Poisson observations," Papers 1904.03356, arXiv.org, revised Mar 2020.
    7. Noba, Kei & Pérez, José-Luis & Yamazaki, Kazutoshi & Yano, Kouji, 2018. "On optimal periodic dividend strategies for Lévy risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 80(C), pages 29-44.
    8. Zbigniew Palmowski & José Luis Pérez & Budhi Arta Surya & Kazutoshi Yamazaki, 2020. "The Leland–Toft optimal capital structure model under Poisson observations," Finance and Stochastics, Springer, vol. 24(4), pages 1035-1082, October.

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