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Discounted penalty function at Parisian ruin for Lévy insurance risk process

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  • Loeffen, R.
  • Palmowski, Z.
  • Surya, B.A.

Abstract

In the setting of a Lévy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold r. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level b), which generalizes known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the q-potential measure of the process killed at Parisian ruin. The results have semi-explicit expressions in terms of the q-scale function and the distribution of the Lévy process.

Suggested Citation

  • Loeffen, R. & Palmowski, Z. & Surya, B.A., 2018. "Discounted penalty function at Parisian ruin for Lévy insurance risk process," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 190-197.
    • Handle: RePEc:eee:insuma:v:83:y:2018:i:c:p:190-197
    DOI: 10.1016/j.insmatheco.2017.10.008
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    References listed on IDEAS

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    1. Dickson,David C. M., 2016. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9781107154605, October.
    2. Mark Broadie & Mikhail Chernov & Suresh Sundaresan, 2007. "Optimal Debt and Equity Values in the Presence of Chapter 7 and Chapter 11," Journal of Finance, American Finance Association, vol. 62(3), pages 1341-1377, June.
    3. 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.
    4. Pascal Francois, 2004. "Capital Structure and Asset Prices: Some Effects of Bankruptcy Procedures," The Journal of Business, University of Chicago Press, vol. 77(2), pages 387-412, April.
    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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    Citations

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    Cited by:

    1. Budhi Surya & Wenyuan Wang & Xianghua Zhao & Xiaowen Zhou, 2020. "Parisian excursion with capital injection for draw-down reflected Levy insurance risk process," Papers 2005.09214, arXiv.org.
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    3. Aili Zhang & Ping Chen & Shuanming Li & Wenyuan Wang, 2020. "Risk Modelling on Liquidations with L\'{e}vy Processes," Papers 2007.01426, arXiv.org.
    4. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    5. Esther Frostig & Adva Keren-Pinhasik, 2020. "Parisian Ruin with Erlang Delay and a Lower Bankruptcy Barrier," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 101-134, March.
    6. Zhang, Aili & Chen, Ping & Li, Shuanming & Wang, Wenyuan, 2022. "Risk modelling on liquidations with Lévy processes," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    7. 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.
    8. Li, Xin & Liu, Haibo & Tang, Qihe & Zhu, Jinxia, 2020. "Liquidation risk in insurance under contemporary regulatory frameworks," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 36-49.
    9. Landriault, David & Li, Bin & Lkabous, Mohamed Amine & Wang, Zijia, 2023. "Bridging the first and last passage times for Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 308-334.
    10. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
    11. Nguyen, Duy Phat & Borovkov, Konstantin, 2023. "Parisian ruin with random deficit-dependent delays for spectrally negative Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 110(C), pages 72-81.
    12. Wang, Wenyuan & Chen, Ping & Li, Shuanming, 2020. "Generalized expected discounted penalty function at general drawdown for Lévy risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 12-25.
    13. Cheung, Eric C.K. & Zhu, Wei, 2023. "Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 84-101.
    14. Eric C. K. Cheung & Jeff T. Y. Wong, 2023. "A Note on a Modified Parisian Ruin Concept," Risks, MDPI, vol. 11(3), pages 1-15, March.
    15. Ran Xu & Wenyuan Wang & Jose Garrido, 2022. "Optimal Dividend Strategy Under Parisian Ruin with Affine Penalty," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1385-1409, September.
    16. Piotr Jaworski & Kamil Liberadzki & Marcin Liberadzki, 2021. "On Write-Down/ Write-Up Loss Absorbing Instruments," European Research Studies Journal, European Research Studies Journal, vol. 0(1), pages 1204-1219.

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    More about this item

    Keywords

    Lévy process; Parisian ruin; Risk process; Ruin; Resolvent; First-passage time;
    All these keywords.

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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