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The discounted penalty function with multi-layer dividend strategy in the phase-type risk model

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  • Jiang, Wuyuan
  • Yang, Zhaojun
  • Li, Xinping

Abstract

This paper considers a Sparre Andersen model in which the inter-claim times have a phase-type distribution and the premium rate is a step function depending on the current surplus level. We derive the system of piecewise integro-differential equations for the Gerber–Shiu discounted penalty functions and obtain the closed form expressions of the Gerber–Shiu functions if the claim amount distribution belongs to the rational family. We provide a recursive approach to calculate Gerber–Shiu functions and present an example.

Suggested Citation

  • Jiang, Wuyuan & Yang, Zhaojun & Li, Xinping, 2012. "The discounted penalty function with multi-layer dividend strategy in the phase-type risk model," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1358-1366.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:7:p:1358-1366
    DOI: 10.1016/j.spl.2012.03.012
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    References listed on IDEAS

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    1. Lin, X. Sheldon & Sendova, Kristina P., 2008. "The compound Poisson risk model with multiple thresholds," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 617-627, April.
    2. Lu, Yi & Li, Shuanming, 2009. "The Markovian regime-switching risk model with a threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 296-303, April.
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    5. Wan, Ning, 2007. "Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 509-523, May.
    6. Lin, X.Sheldon & Pavlova, Kristina P., 2006. "The compound Poisson risk model with a threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 57-80, February.
    7. Shuanming Li, 2008. "The Time of Recovery and the Maximum Severity of Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 12(4), pages 413-425.
    8. Yang, Hu & Zhang, Zhimin, 2008. "Gerber-Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 984-991, June.
    9. Albrecher, Hansjorg & Boxma, Onno J., 2005. "On the discounted penalty function in a Markov-dependent risk model," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 650-672, December.
    10. Jiandong Ren, 2007. "The Discounted Joint Distribution of the Surplus Prior to Ruin and the Deficit at Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(3), pages 128-136.
    11. Li, Shuanming & Lu, Yi, 2008. "The Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-Modulated Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 38(1), pages 53-71, May.
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    Cited by:

    1. 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.
    2. Olena Ragulina & Jonas Šiaulys, 2020. "Upper Bounds and Explicit Formulas for the Ruin Probability in the Risk Model with Stochastic Premiums and a Multi-Layer Dividend Strategy," Mathematics, MDPI, vol. 8(11), pages 1-35, October.
    3. 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.
    4. Wuyuan Jiang & Zhaojun Yang, 2014. "The expected discounted penalty function for two classes of risk processes perturbed by diffusion with multiple thresholds," Indian Journal of Pure and Applied Mathematics, Springer, vol. 45(4), pages 479-495, August.
    5. Zhou, Zhongbao & Xiao, Helu & Deng, Yingchun, 2015. "Markov-dependent risk model with multi-layer dividend strategy," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 273-286.

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