基于Ornstein-Uhlenbeck过程下具有两个再保险公司的比例再保险与投资

数学物理学报 ›› 2023, Vol. 43 ›› Issue (3): 957-969.

基于Ornstein-Uhlenbeck过程下具有两个再保险公司的比例再保险与投资

黄玲¹(),刘海燕^1,²,陈密^1,^2,^*()

¹福建师范大学数学与统计学院福州350117
²福建省分析数学及应用重点实验室福州350117

收稿日期:2022-04-25 修回日期:2023-02-06 出版日期:2023-06-26 发布日期:2023-06-01
通讯作者: 陈密 E-mail:hl2193088930@163.com;chenmi0610@163.com
作者简介:黄玲,E-mail: hl2193088930@163.com
基金资助:
国家自然科学基金(11701087);福建省自然科学基金(2019J01673)

Proportional Reinsurance and Investment Based on the Ornstein-Uhlenbeck Process in the Presence of Two Reinsurers

Huang ##¹(),Liu Haiyan^1,²,Chen Mi^1,^2,^*()

¹School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117
²Fujian Provincial Key Laboratory of Mathematical Analysis and its Applications, Fuzhou 350117

Received:2022-04-25 Revised:2023-02-06 Online:2023-06-26 Published:2023-06-01
Contact: Mi Chen E-mail:hl2193088930@163.com;chenmi0610@163.com
Supported by:
NSFC(11701087);NSF of Fujian Province(2019J01673)

摘要/Abstract

摘要：

该文研究了两类风险模型下具有两个再保险公司的最优再保险和投资问题.保险公司购买比例再保险并投资于无风险资产和风险资产组成的金融市场,其风险资产价格模型受Ornstein-Uhlenbeck过程影响.假设再保险的保费按照指数保费原则来计算,保险公司的目标是使终端财富的期望指数效用最大化.利用随机控制理论和HJB方程,推导出了最优策略和值函数的显式表达式.最后,通过数值分析验证了模型参数对最优策略的影响.

关键词: Ornstein-Uhlenbeck过程, 指数效用, 比例再保险, 投资, 指数保费原则

Abstract:

This paper studies the optimal reinsurance and investment problem with two reinsurance companies under two risk models. The insurance company purchases proportional reinsurance and invests in the financial market consisting of one risk-free asset and one risky asset, where the price of the risky asset is influenced by the Ornstein-Uhlenbeck process. Assuming that premiums for reinsurance are calculated according to the exponential premium principle, and the insurer's goal is to maximize the expected exponential utility of terminal wealth. Using stochastic control theory and HJB equation, the explicit expressions of the optimal strategy and value function are derived. Finally, the influence of model parameters on optimal strategy is verified by numerical analysis.

Key words: Ornstein-Uhlenbeck process, Exponential utility, Proportional reinsurance, Investment, Exponential premium principle

中图分类号:

O211.6

黄玲,刘海燕,陈密. 基于Ornstein-Uhlenbeck过程下具有两个再保险公司的比例再保险与投资[J]. 数学物理学报, 2023, 43(3): 957-969.

Huang ,Liu Haiyan,Chen Mi. Proportional Reinsurance and Investment Based on the Ornstein-Uhlenbeck Process in the Presence of Two Reinsurers[J]. Acta mathematica scientia,Series A, 2023, 43(3): 957-969.

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参考文献 32

[1]	Browne S. Optimal investment policies for a firm with random risk process: exponential utility and minimizing the probability of ruin. Mathematics of Operations Research, 1995, 20(4): 937-958 doi: 10.1287/moor.20.4.937
[2]	Liu C S, Yang H L. Optimal investment for an insurer to minimize its probability of ruin. North American Actuarial Journal, 2004, 8(2): 11-31 doi: 10.1080/10920277.2004.10596134
[3]	Liang Z B. Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion. Acta Mathematicae Applicatae Sinica (English Series), 2007, 23(3): 477-488
[4]	Chen M, Peng X F, Guo J Y. Optimal dividend problem with a nonlinear regular-singular stochastic control. Insurance: Mathematics and Economics, 2013, 52(3): 448-456 doi: 10.1016/j.insmatheco.2013.02.010
[5]	Meng Q B, Li Z D, Wang M H, Zhang X. Stochastic optimal control models for the insurance company with bankruptcy return. Applied Mathematics and Information Sciences, 2013, 7(1L): 273-282
[6]	Li D P, Rong X M, Zhao H. Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model. Journal of Computational and Applied Mathematics, 2015, 283: 142-162 doi: 10.1016/j.cam.2015.01.038
[7]	Xu L, Zhang L M, Yao D J. Optimal investment and reinsurance for an insurer under Markov-modulated financial market. Insurance: Mathematics and Economics, 2017, 74: 7-19 doi: 10.1016/j.insmatheco.2017.02.005
[8]	Wang Y J, Rong X M, Zhao H. Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model. Journal of Computational and Applied Mathematics, 2018, 328: 414-431 doi: 10.1016/j.cam.2017.08.001
[9]	Bi J N, Cai J. Optimal investment-reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets. Insurance: Mathematics and Economics, 2019, 85: 1-14 doi: 10.1016/j.insmatheco.2018.11.007
[10]	Chen M, Yuen K C, Wang W Y. Optimal reinsurance and dividends with transaction costs and taxes under thinning structure. Scandinavian Actuarial Journal, 2021, 2021(3): 198-217 doi: 10.1080/03461238.2020.1824158
[11]	Zhou Z B, Bai Y F, Xiao H L, Chen X. A non-zero-sum reinsurance-investment game with delay and asymmetric information. Journal of Industrial and Management Optimization, 2021, 17(2): 909-936 doi: 10.3934/jimo.2020004
[12]	Li N, Wang W. Optimal dividend and proportional reinsurance strategy under standard deviation premium principle. Bulletin of the Malaysian Mathematical Sciences Society, 2022, 45(2): 869-888 doi: 10.1007/s40840-021-01220-w
[13]	Young V R, Zariphopoulou T. Pricing dynamic insurance risks using the principle of equivalent utility. Scandinavian Actuarial Journal, 2002, 2002(4): 246-279 doi: 10.1080/03461230110106327
[14]	Young V R. Equity-indexed life insurance: pricing and reserving using the principle of equivalent utility. North American Actuarial Journal, 2003, 7(1): 68-86 doi: 10.1080/10920277.2003.10596078
[15]	Moore K S, Young V R. Pricing equity-linked pure endowments via the principle of equivalent utility. Insurance: Mathematics and Economics, 2003, 33(3): 497-516 doi: 10.1016/S0167-6687(03)00166-5
[16]	Musiela M, Zariphopoulou T. An example of indifference prices under exponential preferences. Finance and Stochastics, 2004, 8(2): 229-239 doi: 10.1007/s00780-003-0112-5
[17]	Chen T, Liu W, Tan T, et al. Optimal reinsurance with default risk: a reinsurer's perspective. Journal of Industrial and Management Optimization, 2021, 17(5): 2971-2987 doi: 10.3934/jimo.2020103
[18]	Liang Z B, Yuen K C. Optimal dynamic reinsurance with dependent risks: variance premium principle. Scandinavian Actuarial Journal, 2016, 2016(1): 18-36 doi: 10.1080/03461238.2014.892899
[19]	Han X, Liang Z B, Young V R. Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle. Scandinavian Actuarial Journal, 2020, 2020(10): 879-903 doi: 10.1080/03461238.2020.1788136
[20]	Wen C. Pricing catastrophe reinsurance under the standard deviation premium principle. AIMS Mathematics, 2022, 7(3): 4472-4484 doi: 10.3934/math.2022249
[21]	Chi Y C, Meng H. Optimal reinsurance arrangements in the presence of two reinsurers. Scandinavian Actuarial Journal, 2014, 2014(5): 424-438 doi: 10.1080/03461238.2012.723638
[22]	Chen M, Yuen K C. Optimal dividend and reinsurance in the presence of two reinsurers. Journal of Applied Probability, 2016, 53(2): 554-571 doi: 10.1017/jpr.2016.20
[23]	Meng H, Zhou M, Siu T K. Optimal reinsurance policies with two reinsurers in continuous time. Economic Modelling, 2016, 59(1): 182-195 doi: 10.1016/j.econmod.2016.07.009
[24]	Yao D J, Fan K. Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle. Journal of Industrial and Management Optimization, 2017, 14(3): 1055-1083 doi: 10.3934/jimo.2017090
[25]	Baev A V, Bondarev B V. On the ruin probability of an insurance company dealing in a BS-market. Theory of Probability and Mathematical Statistics, 2007, 74: 11-23
[26]	Liang Z B, Yuen K C, Guo J Y. Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process. Insurance: Mathematics and Economics, 2011, 49: 207-215 doi: 10.1016/j.insmatheco.2011.04.005
[27]	Tian Y X, Guo J Y, Sun Z Y. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial and Management Optimization, 2021, 17(4): 1887-1912 doi: 10.3934/jimo.2020051
[28]	Li Y, Mao X R, Song Y Z, Tao J. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial and Management Optimization, 2022, 18(1): 75-93 doi: 10.3934/jimo.2020143
[29]	A C X, Gu A L, Shao Y. Optimal reinsurance and investment strategy with delay in Heston's SV model. Journal of the Operations Research Society of China, 2021, 9(2): 245-271 doi: 10.1007/s40305-020-00331-8
[30]	Bai Y F, Zhou Z B, Xiao H L, et al. A hybrid stochastic differential reinsurance and investment game with bounded memory. European Journal of Operational Research, 2021, 296(2): 717-737 doi: 10.1016/j.ejor.2021.04.046
[31]	Liu S S, Guo W J, Tong X L. Martingale method for optimal investment and proportional reinsurance. Applied Mathematics: A Journal of Chinese Universities, 2021, 36(1): 16-30 doi: 10.1007/s11766-021-3463-8
[32]	Yang H, Zhang L. Optimal investment for insurer with jump-diffusion risk process. Insurance: Mathematics and Economics, 2005, 37: 615-634 doi: 10.1016/j.insmatheco.2005.06.009