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Stochastic differential portfolio games for an insurer in a jump-diffusion risk process

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  • Xiang Lin

  • Chunhong Zhang

  • Tak Siu

Abstract

We discuss an optimal portfolio selection problem of an insurer who faces model uncertainty in a jump-diffusion risk model using a game theoretic approach. In particular, the optimal portfolio selection problem is formulated as a two-person, zero-sum, stochastic differential game between the insurer and the market. There are two leader-follower games embedded in the game problem: (i) The insurer is the leader of the game and aims to select an optimal portfolio strategy by maximizing the expected utility of the terminal surplus in the “worst-case” scenario; (ii) The market acts as the leader of the game and aims to choose an optimal probability scenario to minimize the maximal expected utility of the terminal surplus. Using techniques of stochastic linear-quadratic control, we obtain closed-form solutions to the game problems in both the jump-diffusion risk process and its diffusion approximation for the case of an exponential utility. Copyright Springer-Verlag 2012

Suggested Citation

  • Xiang Lin & Chunhong Zhang & Tak Siu, 2012. "Stochastic differential portfolio games for an insurer in a jump-diffusion risk process," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 75(1), pages 83-100, February.
    • Handle: RePEc:spr:mathme:v:75:y:2012:i:1:p:83-100
    DOI: 10.1007/s00186-011-0376-z
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    1. Chen, Lv & Shen, Yang & Su, Jianxi, 2020. "A continuous-time theory of reinsurance chains," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 129-146.
    2. Qiang Zhang & Ping Chen, 2020. "Optimal Reinsurance and Investment Strategy for an Insurer in a Model with Delay and Jumps," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 777-801, June.
    3. Feng, Yang & Siu, Tak Kuen & Zhu, Jinxia, 2024. "Optimal payout strategies when Bruno de Finetti meets model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 148-164.
    4. Zeng, Yan & Li, Danping & Gu, Ailing, 2016. "Robust equilibrium reinsurance-investment strategy for a mean–variance insurer in a model with jumps," Insurance: Mathematics and Economics, Elsevier, vol. 66(C), pages 138-152.
    5. Negash Medhin & Chuan Xu, 2020. "Nonzero-Sum Stochastic Differential Reinsurance Games with Jump–Diffusion Processes," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 566-584, November.
    6. Zhao, Hui & Shen, Yang & Zeng, Yan & Zhang, Wenjun, 2019. "Robust equilibrium excess-of-loss reinsurance and CDS investment strategies for a mean–variance insurer with ambiguity aversion," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 159-180.
    7. Peng, Xingchun & Chen, Fenge & Wang, Wenyuan, 2021. "Robust optimal investment and reinsurance for an insurer with inside information," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 15-30.
    8. Ya Huang & Xiangqun Yang & Jieming Zhou, 2017. "Robust optimal investment and reinsurance problem for a general insurance company under Heston model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(2), pages 305-326, April.
    9. Zheng, Xiaoxiao & Zhou, Jieming & Sun, Zhongyang, 2016. "Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 77-87.
    10. A, Chunxiang & Li, Zhongfei, 2015. "Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston’s SV model," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 181-196.
    11. Peng, Xingchun & Chen, Fenge & Hu, Yijun, 2014. "Optimal investment, consumption and proportional reinsurance under model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 222-234.
    12. Chen, Zhiping & Yang, Peng, 2020. "Robust optimal reinsurance–investment strategy with price jumps and correlated claims," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 27-46.
    13. Cheung, Ka Chun & Yam, Sheung Chi Phillip & Zhang, Yiying, 2019. "Risk-adjusted Bowley reinsurance under distorted probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 64-72.
    14. Feng, Yang & Zhu, Jinxia & Siu, Tak Kuen, 2021. "Optimal risk exposure and dividend payout policies under model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 1-29.
    15. Yi, Bo & Li, Zhongfei & Viens, Frederi G. & Zeng, Yan, 2013. "Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 601-614.
    16. Zhu, Huainian & Cao, Ming & Zhang, Chengke, 2019. "Time-consistent investment and reinsurance strategies for mean-variance insurers with relative performance concerns under the Heston model," Finance Research Letters, Elsevier, vol. 30(C), pages 280-291.
    17. Søren Asmussen & Bent Jesper Christensen & Julie Thøgersen, 2019. "Stackelberg Equilibrium Premium Strategies for Push-Pull Competition in a Non-Life Insurance Market with Product Differentiation," Risks, MDPI, vol. 7(2), pages 1-23, May.

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