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Optimal mean–variance reinsurance and investment in a jump-diffusion financial market with common shock dependence

Author

Listed:
  • Zhibin Liang

    (Nanjing Normal University)

  • Junna Bi

    (East China Normal University)

  • Kam Chuen Yuen

    (The University of Hong Kong)

  • Caibin Zhang

    (Nanjing Normal University)

Abstract

In this paper, we study the optimal reinsurance-investment problems in a financial market with jump-diffusion risky asset, where the insurance risk model is modulated by a compound Poisson process, and the two jump number processes are correlated by a common shock. Moreover, we remove the assumption of nonnegativity on the expected value of the jump size in the stock market, which is more economic reasonable since the jump sizes are always negative in the real financial market. Under the criterion of mean–variance, based on the stochastic linear–quadratic control theory, we derive the explicit expressions of the optimal strategies and value function which is a viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. Furthermore, we extend the results in the linear–quadratic setting to the original mean–variance problem, and obtain the solutions of efficient strategy and efficient frontier explicitly. Some numerical examples are given to show the impact of model parameters on the efficient frontier.

Suggested Citation

  • Zhibin Liang & Junna Bi & Kam Chuen Yuen & Caibin Zhang, 2016. "Optimal mean–variance reinsurance and investment in a jump-diffusion financial market with common shock dependence," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 155-181, August.
    • Handle: RePEc:spr:mathme:v:84:y:2016:i:1:d:10.1007_s00186-016-0538-0
    DOI: 10.1007/s00186-016-0538-0
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    References listed on IDEAS

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    1. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    2. Liang, Zhibin & Yuen, Kam Chuen & Guo, Junyi, 2011. "Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 207-215, September.
    3. Junna Bi & Junyi Guo, 2013. "Optimal Mean-Variance Problem with Constrained Controls in a Jump-Diffusion Financial Market for an Insurer," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 252-275, April.
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    6. Liang, Zhibin & Bayraktar, Erhan, 2014. "Optimal reinsurance and investment with unobservable claim size and intensity," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 156-166.
    7. 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.
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    Citations

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    Cited by:

    1. Fudong Wang & Zhibin Liang, 2022. "Optimal Per-Loss Reinsurance for a Risk Model with a Thinning-Dependence Structure," Mathematics, MDPI, vol. 10(23), pages 1-23, December.
    2. Yuan, Yu & Han, Xia & Liang, Zhibin & Yuen, Kam Chuen, 2023. "Optimal reinsurance-investment strategy with thinning dependence and delay factors under mean-variance framework," European Journal of Operational Research, Elsevier, vol. 311(2), pages 581-595.
    3. Caibin Zhang & Zhibin Liang & Kam Chuen Yuen, 2019. "Optimal dynamic reinsurance with common shock dependence and state-dependent risk aversion," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 6(01), pages 1-45, March.
    4. Yu Yuan & Zhibin Liang & Xia Han, 2022. "Minimizing the penalized probability of drawdown for a general insurance company under ambiguity aversion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(2), pages 259-290, October.
    5. Yingxu Tian & Zhongyang Sun & Junyi Guo, 2022. "Optimal Mean-Variance Investment-Reinsurance Strategy for a Dependent Risk Model with Ornstein-Uhlenbeck Process," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1169-1191, June.
    6. Claudia Ceci & Katia Colaneri & Alessandra Cretarola, 2021. "Optimal Reinsurance and Investment under Common Shock Dependence Between Financial and Actuarial Markets," Papers 2105.07524, arXiv.org.
    7. Yanfei Bai & Zhongbao Zhou & Helu Xiao & Rui Gao & Feimin Zhong, 2021. "A stochastic Stackelberg differential reinsurance and investment game with delay in a defaultable market," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 341-381, December.
    8. Yan Zhang & Peibiao Zhao & Rufei Ma, 2022. "Robust Optimal Excess-of-Loss Reinsurance and Investment Problem with more General Dependent Claim Risks and Defaultable Risk," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2743-2777, December.
    9. Yuchen Li & Zongxia Liang & Shunzhi Pang, 2022. "Continuous-Time Monotone Mean-Variance Portfolio Selection," Papers 2211.12168, arXiv.org, revised Jan 2024.
    10. Yingxu Tian & Zhongyang Sun, 2018. "Mean-Variance Portfolio Selection in a Jump-Diffusion Financial Market with Common Shock Dependence," JRFM, MDPI, vol. 11(2), pages 1-12, May.

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