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Stochastic differential reinsurance games with capital injections

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  • Zhang, Nan
  • Jin, Zhuo
  • Qian, Linyi
  • Fan, Kun

Abstract

This paper investigates a class of reinsurance game problems between two insurance companies under the framework of non-zero-sum stochastic differential games. Both insurers can purchase proportional reinsurance contracts from reinsurance markets and have the option of conducting capital injections. We assume the reinsurance premium is calculated under the generalized variance premium principle. The objective of each insurer is to maximize the expected value that synthesizes the discounted utility of his surplus relative to a reference point, the penalties caused by his own capital injection interventions, and the gains brought by capital injections of his competitor. We prove the verification theorem and derive explicit expressions of the Nash equilibrium strategy by solving the corresponding quasi-variational inequalities. Numerical examples are also conducted to illustrate our results.

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  • Zhang, Nan & Jin, Zhuo & Qian, Linyi & Fan, Kun, 2019. "Stochastic differential reinsurance games with capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 7-18.
    • Handle: RePEc:eee:insuma:v:88:y:2019:i:c:p:7-18
    DOI: 10.1016/j.insmatheco.2019.05.002
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    References listed on IDEAS

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    1. René Aïd & Matteo Basei & Giorgia Callegaro & Luciano Campi & Tiziano Vargiolu, 2020. "Nonzero-Sum Stochastic Differential Games with Impulse Controls: A Verification Theorem with Applications," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 205-232, February.
    2. Zhibin Liang & Kam Chuen Yuen & Ka Chun Cheung, 2012. "Optimal reinsurance–investment problem in a constant elasticity of variance stock market for jump‐diffusion risk model," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 28(6), pages 585-597, November.
    3. Li, Peng & Zhou, Ming & Yin, Chuancun, 2015. "Optimal reinsurance with both proportional and fixed costs," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 134-141.
    4. Zhang, Xin & Meng, Hui & Zeng, Yan, 2016. "Optimal investment and reinsurance strategies for insurers with generalized mean–variance premium principle and no-short selling," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 125-132.
    5. Yin, Chuancun & Wen, Yuzhen, 2013. "Optimal dividend problem with a terminal value for spectrally positive Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 769-773.
    6. Løkka, Arne & Zervos, Mihail, 2008. "Optimal dividend and issuance of equity policies in the presence of proportional costs," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 954-961, June.
    7. Borch, Karl, 1960. "Reciprocal Reinsurance Treaties," ASTIN Bulletin, Cambridge University Press, vol. 1(4), pages 170-191, December.
    8. Chuancun Yin & Yuzhen Wen, 2013. "Optimal dividends problem with a terminal value for spectrally positive Levy processes," Papers 1302.6011, arXiv.org.
    9. Wang, Ning & Zhang, Nan & Jin, Zhuo & Qian, Linyi, 2019. "Robust non-zero-sum investment and reinsurance game with default risk," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 115-132.
    10. Bai, Lihua & Guo, Junyi, 2008. "Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 968-975, June.
    11. Bjarne Højgaard & Søren Asmussen & Michael Taksar, 2000. "Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation," Finance and Stochastics, Springer, vol. 4(3), pages 299-324.
    12. Zhou, Ming & Yuen, Kam C., 2012. "Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle," Economic Modelling, Elsevier, vol. 29(2), pages 198-207.
    13. Xin Zhang & Ming Zhou & Junyi Guo, 2007. "Optimal combinational quota‐share and excess‐of‐loss reinsurance policies in a dynamic setting," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 23(1), pages 63-71, January.
    14. 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.
    15. Asmussen, Soren & Taksar, Michael, 1997. "Controlled diffusion models for optimal dividend pay-out," Insurance: Mathematics and Economics, Elsevier, vol. 20(1), pages 1-15, June.
    16. Kristoffer Lindensjo & Filip Lindskog, 2019. "Optimal dividends and capital injection under dividend restrictions," Papers 1902.06294, arXiv.org.
    17. Meng, Hui & Li, Shuanming & Jin, Zhuo, 2015. "A reinsurance game between two insurance companies with nonlinear risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 91-97.
    18. Yao, Dingjun & Yang, Hailiang & Wang, Rongming, 2011. "Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs," European Journal of Operational Research, Elsevier, vol. 211(3), pages 568-576, June.
    19. Hojgaard, Bjarne & Taksar, Michael, 1998. "Optimal proportional reinsurance policies for diffusion models with transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 22(1), pages 41-51, May.
    20. Zhuo Jin & G. Yin, 2013. "Numerical Methods for Optimal Dividend Payment and Investment Strategies of Markov-Modulated Jump Diffusion Models with Regular and Singular Controls," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 246-271, October.
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    Cited by:

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    2. Wang, Ning & Zhang, Nan & Jin, Zhuo & Qian, Linyi, 2021. "Stochastic differential investment and reinsurance games with nonlinear risk processes and VaR constraints," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 168-184.

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