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Optimal Dynamic XL Reinsurance

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  • Hipp, Christian
  • Vogt, Michael

Abstract

We consider a risk process modelled as a compound Poisson process. We find the optimal dynamic unlimited excess of loss reinsurance strategy to minimize infinite time ruin probability, and prove the existence of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation as well as a verification theorem. Numerical examples with exponential, shifted exponential, and Pareto claims are given.

Suggested Citation

  • Hipp, Christian & Vogt, Michael, 2003. "Optimal Dynamic XL Reinsurance," ASTIN Bulletin, Cambridge University Press, vol. 33(2), pages 193-207, November.
    • Handle: RePEc:cup:astinb:v:33:y:2003:i:02:p:193-207_01
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    Cited by:

    1. Stefan Ankirchner & Christophette Blanchet-Scalliet & Nabil Kazi-Tani & Chao Zhou, 2019. "Gambling for resurrection and the heat equation on a triangle," Working Papers hal-02405853, HAL.
    2. Dalila Guerdouh & Nabil Khelfallah & Josep Vives, 2022. "Optimal Control Strategies for the Premium Policy of an Insurance Firm with Jump Diffusion Assets and Stochastic Interest Rate," JRFM, MDPI, vol. 15(3), pages 1-19, March.
    3. Gu, Mengdi & Yang, Yipeng & Li, Shoude & Zhang, Jingyi, 2010. "Constant elasticity of variance model for proportional reinsurance and investment strategies," Insurance: Mathematics and Economics, Elsevier, vol. 46(3), pages 580-587, June.
    4. Tan, Ken Seng & Wei, Pengyu & Wei, Wei & Zhuang, Sheng Chao, 2020. "Optimal dynamic reinsurance policies under a generalized Denneberg’s absolute deviation principle," European Journal of Operational Research, Elsevier, vol. 282(1), pages 345-362.
    5. Thonhauser, Stefan & Albrecher, Hansjorg, 2007. "Dividend maximization under consideration of the time value of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 163-184, July.
    6. Preischl, M. & Thonhauser, S., 2019. "Optimal reinsurance for Gerber–Shiu functions in the Cramér–Lundberg model," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 82-91.
    7. Ekaterina Bulinskaya & Julia Gusak & Anastasia Muromskaya, 2015. "Discrete-time Insurance Model with Capital Injections and Reinsurance," Methodology and Computing in Applied Probability, Springer, vol. 17(4), pages 899-914, December.
    8. Christian Hipp, 2018. "Company Value with Ruin Constraint in Lundberg Models," Risks, MDPI, vol. 6(3), pages 1-15, July.
    9. Chonghu Guan & Zuo Quan Xu & Rui Zhou, 2020. "Dynamic optimal reinsurance and dividend-payout in finite time horizon," Papers 2008.00391, arXiv.org, revised Jun 2022.
    10. Bilel Jarraya & Abdelfettah Bouri, 2013. "A Theoretical Assessment on Optimal Asset Allocations in Insurance Industry," International Journal of Finance & Banking Studies, Center for the Strategic Studies in Business and Finance, vol. 2(4), pages 30-44, October.
    11. J. Cerda-Hernandez & A. Sikov & A. Ramos, 2022. "An optimal investment strategy aimed at maximizing the expected utility across all intermediate capital levels," Papers 2207.02947, arXiv.org, revised Jun 2024.
    12. Nabil Kazi-Tani, 2018. "Inf-Convolution of Choquet Integrals and Applications in Optimal Risk Transfer," Working Papers hal-01742629, HAL.
    13. Cerqueti, Roy & Foschi, Rachele & Spizzichino, Fabio, 2009. "A spatial mixed Poisson framework for combination of excess-of-loss and proportional reinsurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 59-64, August.
    14. Xue, Xiaole & Wei, Pengyu & Weng, Chengguo, 2019. "Derivatives trading for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 40-53.
    15. Irgens, Christian & Paulsen, Jostein, 2004. "Optimal control of risk exposure, reinsurance and investments for insurance portfolios," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 21-51, August.
    16. Stefan Ankirchner & Christophette Blanchet-Scalliet & Nabil Kazi-Tani & Chao Zhou, 2021. "Gambling for resurrection and the heat equation on a triangle," Post-Print hal-02405853, HAL.
    17. Schmidli, Hanspeter, 2005. "On optimal investment and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 36(1), pages 25-35, February.
    18. Arian Cani & Stefan Thonhauser, 2017. "An optimal reinsurance problem in the Cramér–Lundberg model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(2), pages 179-205, April.
    19. Hipp, Christian & Taksar, Michael, 2010. "Optimal non-proportional reinsurance control," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 246-254, October.
    20. Anna Castañer & M. Claramunt & Maite Mármol, 2012. "Ruin probability and time of ruin with a proportional reinsurance threshold strategy," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 614-638, October.
    21. Yan Tong & Tongling Lv & Yu Yan, 2023. "Optimal Investment and Reinsurance Policies in a Continuous-Time Model," Mathematics, MDPI, vol. 11(24), pages 1-20, December.
    22. Chen, Shumin & Liu, Yanchu & Weng, Chengguo, 2019. "Dynamic risk-sharing game and reinsurance contract design," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 216-231.

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