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Gerber-Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy

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  • Yang, Hu
  • Zhang, Zhimin

Abstract

This paper studies a Sparre Andersen model in which the inter-claim times are generalized Erlang(n) distributed. We assume that the premium rate is a step function depending on the current surplus level. A piecewise integro-differential equation for the Gerber-Shiu discounted penalty function is derived and solved. Finally, to illustrate the solution procedure, explicit expression for the Laplace transform of the time to ruin is given when the inter-claim times are generalized Erlang(2) distributed and the claim amounts are exponentially distributed.

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  • 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.
    • Handle: RePEc:eee:insuma:v:42:y:2008:i:3:p:984-991
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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.
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    Cited by:

    1. 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.
    2. Yujuan Huang & Wenguang Yu, 2013. "Studies on a Double Poisson-Geometric Insurance Risk Model with Interference," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-8, April.
    3. Yang, Hu & Zhang, Zhimin, 2009. "The perturbed compound Poisson risk model with multi-layer dividend strategy," Statistics & Probability Letters, Elsevier, vol. 79(1), pages 70-78, January.
    4. 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.
    5. Mitric, Ilie-Radu & Sendova, Kristina P. & Tsai, Cary Chi-Liang, 2010. "On a multi-threshold compound Poisson process perturbed by diffusion," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 366-375, March.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Deng, Chao & Zhou, Jieming & Deng, Yingchun, 2012. "The Gerber–Shiu discounted penalty function in a delayed renewal risk model with multi-layer dividend strategy," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1648-1656.
    11. 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.
    12. Shi, Yafeng & Liu, Peng & Zhang, Chunsheng, 2013. "On the compound Poisson risk model with dependence and a threshold dividend strategy," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 1998-2006.
    13. Apostolos D. Papaioannou & Lewis Ramsden, 2022. "Recursive Approaches for Multi-Layer Dividend Strategies in a Phase-Type Renewal Risk Model," Risks, MDPI, vol. 11(1), pages 1-21, December.

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