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Minimizing the Ruin Probability under the Sparre Andersen Model

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  • Linlin Tian
  • Lihua Bai

Abstract

In this paper, we consider the problem of minimizing the ruin probability of an insurance company in which the surplus process follows the Sparre Andersen model. Similar to Bai et al. \cite{bai2017optimal}, we recast this problem in a Markovian framework by adding another dimension representing the time elapsed since the last claim. After Markovization, We investigate the regularity properties of the value function and state the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution to the associated Hamilton-Jacobi-Bellman equation. It should be noted that there is no discount factor in our paper, which makes it tricky to prove the uniqueness. To overcome this difficulty, we construct the strict viscosity supersolution. Then instead of comparing the usual viscosity supersolution and subsolution, we compare the supersolution and the strict subsolution. Eventually we show that all viscosity subsolution is less than the supersolution.

Suggested Citation

  • Linlin Tian & Lihua Bai, 2020. "Minimizing the Ruin Probability under the Sparre Andersen Model," Papers 2004.08124, arXiv.org.
    • Handle: RePEc:arx:papers:2004.08124
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    References listed on IDEAS

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    1. Pablo Azcue & Nora Muler, 2013. "Minimizing the ruin probability allowing investments in two assets: a two-dimensional problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(2), pages 177-206, April.
    2. Liang, Xiaoqing & Young, Virginia R., 2018. "Minimizing the probability of ruin: Optimal per-loss reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 181-190.
    3. Meng, Hui & Zhang, Xin, 2010. "Optimal Risk Control for The Excess of Loss Reinsurance Policies," ASTIN Bulletin, Cambridge University Press, vol. 40(1), pages 179-197, May.
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    5. Pablo Azcue & Nora Muler, 2005. "Optimal Reinsurance And Dividend Distribution Policies In The Cramér‐Lundberg Model," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 261-308, April.
    6. Lesław Gajek & Dariusz Zagrodny, 2004. "Reinsurance Arrangements Maximizing Insurer's Survival Probability," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 71(3), pages 421-435, September.
    7. Hipp, Christian & Plum, Michael, 2000. "Optimal investment for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 27(2), pages 215-228, October.
    8. Li, Danping & Young, Virginia R., 2019. "Optimal reinsurance to minimize the discounted probability of ruin under ambiguity," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 143-152.
    9. Hipp, Christian & Taksar, Michael, 2010. "Optimal non-proportional reinsurance control," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 246-254, October.
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