均值-方差准则下再保险双方联合收益的最优投资再保险策略
Optimal Investment Reinsurance Strategy for the Joint Benefits of the Insurer and the Reinsurer under the Mean-Variance
摘要: 在两类保险业务相关的风险模型背景下，研究了保险公司和再保险公司的联合收益的最优投资再保险问题。假设保险公司可以向再保险公司购买比例再保险和投资于一个由无风险资产和风险资产组成的金融市场，再保险公司采用期望保费原理收取保费并可以通过投资无风险资产来降低风险。 在均值-方差准则下，通过求解扩展的Hamilton-Jacobi-Bellman方程组，得到了联合收益的最优投资策略和最优再保险策略的表达式以及最优值函数，并通过实例验证了结果的有效性。
Abstract: Under the background of the risk model related to two types of insurance business, the optimal investment reinsurance of the joint benefits of the insurer and the reinsurer is studied. Assuming that the insurer can buy proportional reinsurance from the reinsurer and invest in a financial market consisting of risk-free and risk assets, the reinsurer can use the expected premium principle to charge premiums and reduce risk by investing in risk-free assets. Under the mean-variance criterion, the expressions of the optimal investment strategy and the optimal reinsurance strategy and the optimal value function of the combined returns are obtained by solving the extended Hamilton- Jacobi-Bellman equation system, and the validity of the result is verified by example.
文章引用：孙婷婷, 王慧慧, 舒慧生. 均值-方差准则下再保险双方联合收益的最优投资再保险策略[J]. 理论数学, 2021, 11(11): 1850-1870. https://doi.org/10.12677/PM.2021.1111208

参考文献

[1] Hipp, C. and Plum, M. (2000) Optimal Investment for Insurers. Insurance Mathematics and Economics, 27, 215-228.
https://doi.org/10.1016/S0167-6687(00)00049-4
[2] Schmidli, H. (2002) On Minimizing the Ruin Probability by Investment and Reinsurance. Annals of Applied Probability, 12, 890-907.
https://doi.org/10.1214/aoap/1031863173
[3] Promislow, S.D. and Young, V.R. (2005) Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift. North American Actuarial Journal, 9, 110-128.
https://doi.org/10.1080/10920277.2005.10596214
[4] Chen, S., Li, Z. and Li, K. (2010) Optimal Investment-Reinsurance Policy for an Insurance Company with VaR Constraint. Insurance Mathematics and Economics, 47, 144-153.
https://doi.org/10.1016/j.insmatheco.2010.06.002
[5] Browne, S. (1995) Optimal Investment Policies for a Firm with a Random Risk Process: Expo- nential Utility and Minimizing the Probability of Ruin. Mathematics of Operations Research, 20, 937-958.
https://doi.org/10.1287/moor.20.4.937
[6] Bai, L. and Guo, J. (2008) Optimal Proportional Reinsurance and Investment with Multiple Risky Assets and No-Shorting Constraint. Insurance Mathematics Economics, 42, 968-975.
https://doi.org/10.1016/j.insmatheco.2007.11.002
[7] Gu, A., Guo, X., Li, Z., et al. (2012) Optimal Control of Excess-of-Loss Reinsurance and Investment for Insurers under a CEV Model. Insurance: Mathematics and Economics, 51, 674-684.
https://doi.org/10.1016/j.insmatheco.2012.09.003
[8] Markowitz, H.M. (1952) Portfolio Selection. The Journal of Finance, 7, 77-91.
https://doi.org/10.2307/2975974
[9] Zeng, Y. and Li, Z. (2011) Optimal Time-Consistent Investment and Reinsurance Policies for Mean-Variance Insurers. Insurance Mathematics Economics, 49, 145-154.
https://doi.org/10.1016/j.insmatheco.2011.01.001
[10] Bi, J. and Guo, J. (2013) Optimal Mean-Variance Problem with Constrained Controls in a Jump-Diffusion Financial Market for an Insurer. Journal of Optimization Theory Applications, 157, 252-275.
https://doi.org/10.1007/s10957-012-0138-y
[11] Borch, K. (1969) The Optimal Reinsurance Treaty. ASTIN Bulletin, 5, 293-297.
https://doi.org/10.1017/S051503610000814X
[12] Cai, J., Fang, Y., Li, Z., et al. (2013) Optimal Reciprocal Reinsurance Treaties under the Joint Survival Probability and the Joint Profitable Probability. Journal of Risk and Insurance, 80, 145-168.
https://doi.org/10.1111/j.1539-6975.2012.01462.x
[13] Fang, Y. and Qu, Z. (2014) Optimal Combination of Quota-Share and Stop-Loss Reinsurance Treaties under the Joint Survival Probability. IMA Journal of Management Mathematics, 25, 89-103.
https://doi.org/10.1093/imaman/dps029
[14] Li, D., Rong, X. and Zhao, H. (2014) Optimal Reinsurance-Investment Problem for Maximizing the Product of the Insurer’s and the Reinsurer’s Utilities under a CEV Model. Journal of Computational and Applied Mathematics, 255, 671-683.
https://doi.org/10.1016/j.cam.2013.06.033
[15] Li, D., Rong, X. and Zhao, H. (2015) Time-Consistent Reinsurance-Investment Strategy for an Insurer and a Reinsurer with Mean-Variance Criterion under the CEV Model. Journal of Computational Applied Mathematics, 283, 142-162.
https://doi.org/10.1016/j.cam.2015.01.038
[16] Li, D., Rong, X. and Zhao, H. (2016) Optimal Reinsurance and Investment Problem for an Insurer and a Reinsurer with Jump-Diffusion Risk Process under the Heston Model. Compu- tational and Applied Mathematics, 35, 533-557.
https://doi.org/10.1007/s40314-014-0204-1
[17] Hu, H., Yin, Z. and Gao, X. (2018) Optimal Reinsurance-Investment Problem for an Insurer and a Reinsurer with Jump-Diffusion Process. Discrete Dynamics in Nature and Society, 2018, Article ID: 9424908.
https://doi.org/10.1155/2018/9424908
[18] Chen, L. and Shen, Y. (2019) Stochastic Stackelberg Differential Reinsurance Games under Time-Inconsistent Mean-Variance Framework. Insurance: Mathematics and Economics, 88, 120-137.
https://doi.org/10.1016/j.insmatheco.2019.06.006
[19] Bai, Y., Zhou, Z., Gao, R., et al. (2020) Nash Equilibrium Investment-Reinsurance Strate- gies for an Insurer and a Reinsurer with Intertemporal Restrictions and Common Interests. Mathematics, 8, 139.
https://doi.org/10.3390/math8010139
[20] Liang, Z. and Yuen, K.C. (2016) Optimal Dynamic Reinsurance with Dependent Risks: Vari- ance Premium Principle. Scandinavian Actuarial Journal, 2016, 18-36.
https://doi.org/10.1080/03461238.2014.892899
[21] Liang, Z., Bi, J. and Yuen, K.C., et al. (2016) Optimal Mean-Variance Reinsurance and Invest- ment in a Jump-Diffusion Financial Market with Common Shock Dependence. Mathematical Methods of Operations Research, 84, 155-181.
https://doi.org/10.1007/s00186-016-0538-0
[22] Bjork, T. and Murgoci, A. (2010) A General Theory of Markovian Time Inconsistent Stochastic Control Problems. SSRN Electronic Journal.

为你推荐