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Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims

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  • Cheung, Eric C.K.
  • Zhu, Wei

Abstract

The Parisian ruin time, which is the first time the insurer's surplus process has an excursion below level zero that exceeds a prescribed time length, has been extensively analyzed in recent years mainly in the Lévy model and its special cases. However, the cumulative Parisian ruin time, which is the first time the total time spent by the surplus process below level zero exceeds a certain time length, has been rarely considered in the literature. In this paper, we study the cumulative Parisian ruin problem in a renewal risk model with general interclaim times and exponential claims. Explicit formulas for the infinite-time cumulative Parisian ruin probability is first derived under a deterministic Parisian clock and then under an Erlang clock, where the latter case can also serve as an approximation of the former. The finite-time cumulative Parisian ruin probability is subsequently analyzed as well when the time horizon is another Erlang random variable. Our formulas are applied in various numerical examples where the interclaim times follow gamma, Weibull, or Pareto distribution. Consequently, we demonstrate that the choice of the interclaim distribution does have a significant impact on the cumulative Parisian ruin probabilities when one deviates from the exponential assumption.

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  • 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.
    • Handle: RePEc:eee:insuma:v:111:y:2023:i:c:p:84-101
    DOI: 10.1016/j.insmatheco.2023.03.003
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    1. Willmot, Gordon E., 2007. "On the discounted penalty function in the renewal risk model with general interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 17-31, July.
    2. Li, Shu & Zhou, Xiaowen, 2022. "The Parisian and ultimate drawdowns of Lévy insurance models," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 140-160.
    3. Landriault, David & Shi, Tianxiang & Willmot, Gordon E., 2011. "Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 371-379.
    4. Carr, Peter, 1998. "Randomization and the American Put," The Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 597-626.
    5. Frostig, Esther & Pitts, Susan M. & Politis, Konstadinos, 2012. "The time to ruin and the number of claims until ruin for phase-type claims," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 19-25.
    6. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    7. 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.
    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. 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.
    10. Landriault, David & Li, Bin & Shi, Tianxiang & Xu, Di, 2019. "On the distribution of classic and some exotic ruin times," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 38-45.
    11. Lkabous, Mohamed Amine & Czarna, Irmina & Renaud, Jean-François, 2017. "Parisian ruin for a refracted Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 153-163.
    12. Asmussen, Soren & Avram, Florin & Usabel, Miguel, 2002. "Erlangian Approximations for Finite-Horizon Ruin Probabilities," ASTIN Bulletin, Cambridge University Press, vol. 32(2), pages 267-281, November.
    13. Landriault, David & Willmot, Gordon, 2008. "On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 600-608, April.
    14. Eric C. K. Cheung & Zhimin Zhang, 2019. "Periodic threshold-type dividend strategy in the compound Poisson risk model," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(1), pages 1-31, January.
    15. Mogens Bladt & Bo Friis Nielsen & Oscar Peralta, 2019. "Parisian types of ruin probabilities for a class of dependent risk-reserve processes," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(1), pages 32-61, January.
    16. Borovkov, Konstantin A. & Dickson, David C.M., 2008. "On the ruin time distribution for a Sparre Andersen process with exponential claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1104-1108, June.
    17. Mohamed Amine Lkabous & Jean-François Renaud, 2018. "A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model," Risks, MDPI, vol. 6(3), pages 1-11, August.
    18. Antill, Samuel & Grenadier, Steven R., 2019. "Optimal capital structure and bankruptcy choice: Dynamic bargaining versus liquidation," Journal of Financial Economics, Elsevier, vol. 133(1), pages 198-224.
    19. 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.
    20. Guérin, Hélène & Renaud, Jean-François, 2017. "On the distribution of cumulative Parisian ruin," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 116-123.
    21. Bin Li & Qihe Tang & Lihe Wang & Xiaowen Zhou, 2014. "Liquidation risk in the presence of Chapters 7 and 11 of the US bankruptcy code," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(03), pages 1-19.
    22. 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.
    23. Irmina Czarna, 2016. "Parisian ruin probability with a lower ultimate bankrupt barrier," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2016(4), pages 319-337, April.
    24. 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.
    25. V. Ramaswami & Douglas Woolford & David Stanford, 2008. "The erlangization method for Markovian fluid flows," Annals of Operations Research, Springer, vol. 160(1), pages 215-225, April.
    26. Stanford, D.A. & Avram, F. & Badescu, A.L. & Breuer, L. & Silva Soares, A. Da & Latouche, G., 2005. "Phase-type Approximations to Finite-time Ruin Probabilities in the Sparre-Andersen and Stationary Renewal Risk Models," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 131-144, May.
    27. Cheung, Eric C.K. & Landriault, David & Willmot, Gordon E. & Woo, Jae-Kyung, 2010. "Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 117-126, February.
    28. Dickson, David C.M. & Willmot, Gordon E., 2005. "The Density of the Time to Ruin in the Classical Poisson Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 45-60, May.
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    More about this item

    Keywords

    Cumulative Parisian ruin; Occupation time; Renewal risk model; Finite-time ruin probability; Erlangization;
    All these keywords.

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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