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Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods

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  • Jin, Zhuo
  • Yin, G.
  • Wu, Fuke

Abstract

This work develops a stochastic differential game model between two insurance companies who adopt the optimal reinsurance strategies to reduce the risk. The surplus is modeled by a regime-switching jump diffusion process. A single payoff function is imposed, and one player devises an optimal strategy to maximize the expected payoff function, whereas the other player is trying to minimize the same quantity. Using dynamic programming principle, the upper and lower values of the game satisfy a coupled system of nonlinear integro-differential Hamilton–Jacobi–Isaacs (HJI) equations. Moreover, the existence of the saddle point for this game problem is verified. Because of the jumps and regime-switching, closed-form solutions are virtually impossible to obtain. Our effort is devoted to designing numerical methods. We use Markov chain approximation techniques to construct a discrete-time controlled Markov chain to approximate the value functions and optimal controls. Convergence of the approximation algorithms is proved. Examples are presented to illustrate the applicability of the numerical methods.

Suggested Citation

  • Jin, Zhuo & Yin, G. & Wu, Fuke, 2013. "Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 733-746.
    • Handle: RePEc:eee:insuma:v:53:y:2013:i:3:p:733-746
    DOI: 10.1016/j.insmatheco.2013.09.015
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    References listed on IDEAS

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    3. Taksar, Michael & Zeng, Xudong, 2011. "Optimal non-proportional reinsurance control and stochastic differential games," Insurance: Mathematics and Economics, Elsevier, vol. 48(1), pages 64-71, January.
    4. Sotomayor, Luz R. & Cadenillas, Abel, 2011. "Classical and singular stochastic control for the optimal dividend policy when there is regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 344-354, May.
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    Cited by:

    1. Ning Bin & Huainian Zhu & Chengke Zhang, 2023. "Stochastic Differential Games on Optimal Investment and Reinsurance Strategy with Delay Under the CEV Model," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-27, June.
    2. Zou, Bin & Cadenillas, Abel, 2014. "Optimal investment and risk control policies for an insurer: Expected utility maximization," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 57-67.
    3. Asmussen, Søren & Christensen, Bent Jesper & Thøgersen, Julie, 2019. "Nash equilibrium premium strategies for push–pull competition in a frictional non-life insurance market," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 92-100.
    4. Bin Zou & Abel Cadenillas, 2017. "Optimal Investment and Liability Ratio Policies in a Multidimensional Regime Switching Model," Risks, MDPI, vol. 5(1), pages 1-22, January.
    5. Jin, Zhuo & Yang, Hailiang & Yin, G., 2021. "A hybrid deep learning method for optimal insurance strategies: Algorithms and convergence analysis," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 262-275.
    6. 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.
    7. Jin, Zhuo & Yang, Hailiang & Yin, G., 2015. "Optimal debt ratio and dividend payment strategies with reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 351-363.
    8. Cao, Jingyi & Li, Dongchen & Young, Virginia R. & Zou, Bin, 2023. "Reinsurance games with two reinsurers: Tree versus chain," European Journal of Operational Research, Elsevier, vol. 310(2), pages 928-941.
    9. Wei-han Liu, 2023. "Attaining stochastic optimal control over debt ratios in U.S. markets," Review of Quantitative Finance and Accounting, Springer, vol. 61(3), pages 967-993, October.
    10. Liu, Guo & Jin, Zhuo & Li, Shuanming & Zhang, Jiannan, 2022. "Stochastic asset allocation and reinsurance game under contagious claims," Finance Research Letters, Elsevier, vol. 49(C).
    11. 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.
    12. 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.
    13. Claire Mouminoux & Christophe Dutang & Stéphane Loisel & Hansjoerg Albrecher, 2022. "On a Markovian Game Model for Competitive Insurance Pricing," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1061-1091, June.
    14. Chen, Lv & Qian, Linyi & Shen, Yang & Wang, Wei, 2016. "Constrained investment–reinsurance optimization with regime switching under variance premium principle," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 253-267.
    15. Y. Zhang & Z. Jin & J. Wei & G. Yin, 2022. "Mean-variance portfolio selection with dynamic attention behavior in a hidden Markov model," Papers 2205.08743, arXiv.org.
    16. Bin Zou & Abel Cadenillas, 2014. "Optimal Investment and Risk Control Problem for an Insurer: Expected Utility Maximization," Papers 1402.3560, arXiv.org, revised Mar 2014.
    17. 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.
    18. 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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