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Generalized expected discounted penalty function at general drawdown for Lévy risk processes

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  • Wang, Wenyuan
  • Chen, Ping
  • Li, Shuanming

Abstract

This paper considers an insurance surplus process modeled by a spectrally negative Lévy process. Instead of the time of ruin in the traditional setting, we apply the time of drawdown as the risk indicator in this paper. We study the joint distribution of the time of drawdown, the running maximum at drawdown, the last minimum before drawdown, the surplus before drawdown and the surplus at drawdown (may not be deficit in this case), which generalizes the known results on the classical expected discounted penalty function in Gerber and Shiu (1998). The results have semi-explicit expressions in terms of the q-scale functions and the Lévy measure associated with the Lévy process. As applications, the obtained result is applied to recover results in the literature and to obtain new results for the Gerber–Shiu function at ruin for risk processes embedded with a loss-carry-forward taxation system or a barrier dividend strategy. Moreover, numerical examples are provided to illustrate the results.

Suggested Citation

  • Wang, Wenyuan & Chen, Ping & Li, Shuanming, 2020. "Generalized expected discounted penalty function at general drawdown for Lévy risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 12-25.
    • Handle: RePEc:eee:insuma:v:91:y:2020:i:c:p:12-25
    DOI: 10.1016/j.insmatheco.2019.12.002
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    References listed on IDEAS

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    Cited by:

    1. Aili Zhang & Ping Chen & Shuanming Li & Wenyuan Wang, 2020. "Risk Modelling on Liquidations with L\'{e}vy Processes," Papers 2007.01426, arXiv.org.
    2. 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.
    3. Zhang, Aili & Chen, Ping & Li, Shuanming & Wang, Wenyuan, 2022. "Risk modelling on liquidations with Lévy processes," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    4. Kang Hu & Ya Huang & Yingchun Deng, 2023. "Estimating the Gerber–Shiu Function in the Two-Sided Jumps Risk Model by Laguerre Series Expansion," Mathematics, MDPI, vol. 11(9), pages 1-30, April.
    5. 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.
    6. Leonie Violetta Brinker, 2021. "Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model," Risks, MDPI, vol. 9(1), pages 1-18, January.

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