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Optimal Mean-Variance Problem with Constrained Controls in a Jump-Diffusion Financial Market for an Insurer

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  • Junna Bi

    (East China Normal University)

  • Junyi Guo

    (Nankai University)

Abstract

In this paper, we study the optimal investment and optimal reinsurance problem for an insurer under the criterion of mean-variance. The insurer’s risk process is modeled by a compound Poisson process and the insurer can invest in a risk-free asset and a risky asset whose price follows a jump-diffusion process. In addition, the insurer can purchase new business (such as reinsurance). The controls (investment and reinsurance strategies) are constrained to take nonnegative values due to nonnegative new business and no-shorting constraint of the risky asset. We use the stochastic linear-quadratic (LQ) control theory to derive the optimal value and the optimal strategy. The corresponding Hamilton–Jacobi–Bellman (HJB) equation no longer has a classical solution. With the framework of viscosity solution, we give a new verification theorem, and then the efficient strategy (optimal investment strategy and optimal reinsurance strategy) and the efficient frontier are derived explicitly.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:157:y:2013:i:1:d:10.1007_s10957-012-0138-y
    DOI: 10.1007/s10957-012-0138-y
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. Bi, Junna & Cai, Jun, 2019. "Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 1-14.
    5. 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.
    6. 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.
    7. Jiaqi Zhu & Shenghong Li, 2020. "Time-Consistent Investment and Reinsurance Strategies for Mean-Variance Insurers under Stochastic Interest Rate and Stochastic Volatility," Mathematics, MDPI, vol. 8(12), pages 1-22, December.
    8. Yuchen Li & Zongxia Liang & Shunzhi Pang, 2022. "Continuous-Time Monotone Mean-Variance Portfolio Selection," Papers 2211.12168, arXiv.org, revised Jan 2024.
    9. Zhongyang Sun & Junyi Guo, 2018. "Optimal mean–variance investment and reinsurance problem for an insurer with stochastic volatility," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(1), pages 59-79, August.
    10. Junna Bi & Zhibin Liang & Kam Chuen Yuen, 2019. "Optimal mean–variance investment/reinsurance with common shock in a regime-switching market," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(1), pages 109-135, August.
    11. Bi, Junna & Liang, Zhibin & Xu, Fangjun, 2016. "Optimal mean–variance investment and reinsurance problems for the risk model with common shock dependence," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 245-258.
    12. 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.
    13. 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.
    14. Zilan Liu & Yijun Wang & Ya Huang & Jieming Zhou, 2022. "Optimal Time-Consistent Investment and Premium Control Strategies for Insurers with Constraint under the Heston Model," Mathematics, MDPI, vol. 10(7), pages 1-22, March.

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