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The Gerber–Shiu discounted penalty function in a delayed renewal risk model with multi-layer dividend strategy

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  • Deng, Chao
  • Zhou, Jieming
  • Deng, Yingchun

Abstract

A class of delayed renewal risk processes with multi-layer dividend strategy is addressed here. Under the assumption 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 in the delayed renewal risk model is derived, as an analogue of that in the ordinary renewal model, and the relationship between this function and the one in the ordinary renewal model is investigated. Subsequently, this relationship is detailed as regards both the stationary renewal risk model and the ruin probability. Finally, explicit expressions for some ruin-related quantities are included to illustrate the procedure, where the inter-claim times are generalized Erlang(2) distributed and the claims are exponentially distributed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:9:p:1648-1656
    DOI: 10.1016/j.spl.2012.05.002
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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. Willmot, Gordon E. & Dickson, David C. M., 2003. "The Gerber-Shiu discounted penalty function in the stationary renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 403-411, July.
    3. 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.
    4. 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.
    5. 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.
    6. Willmot, Gordon E., 2004. "A note on a class of delayed renewal risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 34(2), pages 251-257, April.
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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.

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