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Markov-dependent risk model with multi-layer dividend strategy

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  • Zhou, Zhongbao
  • Xiao, Helu
  • Deng, Yingchun

Abstract

In this paper, we propose a Markov-dependent risk model with multi-layer dividend strategy in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain. A system of piecewise integro-differential equations with boundary conditions satisfied by the Gerber–Shiu function, with given initial environment state, is derived. The closed form expression of the Gerber–Shiu function is obtained when all the claim amount distributions belong to the rational family. Also, we adopt the Chebyshev polynomial approach to find the approximate solution of the integro-differential equation. The numerical simulation results show the effectiveness of the proposed methods.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:252:y:2015:i:c:p:273-286
    DOI: 10.1016/j.amc.2014.12.016
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    References listed on IDEAS

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    1. Chen, Mi & Yuen, Kam Chuen & Guo, Junyi, 2014. "Survival probabilities in a discrete semi-Markov risk model," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 205-215.
    2. Hansjörg Albrecher & Jürgen Hartinger, 2007. "A Risk Model with Multilayer Dividend Strategy," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 43-64.
    3. 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.
    4. Zhimin Zhang & Hailiang Yang & Hu Yang, 2012. "On a Sparre Andersen Risk Model with Time-Dependent Claim Sizes and Jump-Diffusion Perturbation," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 973-995, December.
    5. Chen, Xu & Xiao, Ting & Yang, Xiang-qun, 2014. "A Markov-modulated jump-diffusion risk model with randomized observation periods and threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 76-83.
    6. Diko, Peter & Usábel, Miguel, 2011. "A numerical method for the expected penalty-reward function in a Markov-modulated jump-diffusion process," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 126-131, July.
    7. Dickson, David C. M. & Hipp, Christian, 2001. "On the time to ruin for Erlang(2) risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 333-344, December.
    8. Meng, Qingbin & Zhang, Xin & Guo, Junyi, 2008. "On a risk model with dependence between claim sizes and claim intervals," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1727-1734, September.
    9. Hans Gerber & Elias Shiu, 2005. "The Time Value of Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 49-69.
    10. 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.
    11. 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.
    12. Janssen, Jacques & Reinhard, Jean-Marie, 1985. "Probabilités de Ruine pour une Classe de Modèles de Risque Semi-Markoviens," ASTIN Bulletin, Cambridge University Press, vol. 15(2), pages 123-133, November.
    13. 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.
    14. Eric Cheung & David Landriault, 2009. "Analysis of a Generalized Penalty Function in a Semi-Markovian Risk Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(4), pages 497-513.
    15. Li, Shuanming & Garrido, Jose, 2004. "On ruin for the Erlang(n) risk process," Insurance: Mathematics and Economics, Elsevier, vol. 34(3), pages 391-408, June.
    16. 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.
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    Cited by:

    1. 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.
    2. Liu, Zhang & Chen, Ping & Hu, Yijun, 2020. "On the dual risk model with diffusion under a mixed dividend strategy," Applied Mathematics and Computation, Elsevier, vol. 376(C).
    3. 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.
    4. 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.

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