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The Gerber–Shiu discounted penalty function in the classical risk model with impulsive dividend policy

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  • Liu, Xiangdong
  • Xiong, Jie
  • Zhang, Shuaiqi

Abstract

In this paper, we study the Gerber–Shiu discounted penalty function in the classical risk model with impulsive dividends. When the surplus process hits a barrier b, the dividend will be paid and the surplus is reduced to a level a. An integro-differential equation for the Gerber–Shiu discounted penalty function is derived by analyzing the evolution of the surplus process and it is solved by Dickson–Hipp operator method. For this process, we also investigate the Laplace transform of the time of ruin, the distribution of the surplus immediately before ruin and the deficit at ruin. These quantities for the special case where the claim size is exponentially distributed are obtained explicitly. Moreover, the distribution of the number of dividends is derived.

Suggested Citation

  • Liu, Xiangdong & Xiong, Jie & Zhang, Shuaiqi, 2015. "The Gerber–Shiu discounted penalty function in the classical risk model with impulsive dividend policy," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 183-190.
    • Handle: RePEc:eee:stapro:v:107:y:2015:i:c:p:183-190
    DOI: 10.1016/j.spl.2015.08.025
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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. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    3. Yuen, Kam C. & Wang, Guojing & Li, Wai K., 2007. "The Gerber-Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 104-112, January.
    4. 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.
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    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. Sheldon Lin, X. & E. Willmot, Gordon & Drekic, Steve, 2003. "The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 551-566, December.
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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. 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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