# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022130
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## Non-zero-sum reinsurance and investment game with correlation between insurance market and financial market under CEV model

 1 School of Mathematics, Tianjin University, 135 Yaguan Road, Tianjin 300354, China 2 School of Mathematics, Center for Applied Mathematics, Tianjin University

*Corresponding author: Hui Zhao

Received  October 2021 Revised  March 2022 Early access August 2022

Fund Project: The first author is supported by the National Natural Science Foundation of China (Grant Nos. 11871052, 11771329, 12171360), Key Project of the Nation Social Science Foundation of China (Grant No. 21AZD071) and Natural Science Foundation of Tianjin City (Grant No. 20JCYBJC01160)

In this paper, we study a non-zero-sum investment and reinsurance game for two insurers. Each insurer's surplus process is described by a Brownian motion with drift. Both insurers are allowed to purchase proportional reinsurance and invest in a risk-free asset and a risky asset. The price process of the risky asset follows the constant elasticity of variance (CEV) model, and the correlation between the risky asset's price process and the claim process is considered. Each insurer aims to maximize the expected exponential utility of his terminal wealth relative to that of his competitor. By applying stochastic control theory, we establish the corresponding Hamilton-Jacobi-Bellman (HJB) equation and derive optimal investment-reinsurance strategies for two insurers under exponential utility function. Furthermore, we consider the insurer's optimal investment-reinsurance strategies without competition. Finally, numerical analyses are provided to analyze the effects of model parameters on the optimal strategies.

Citation: Xue Dong, Ximin Rong, Hui Zhao. Non-zero-sum reinsurance and investment game with correlation between insurance market and financial market under CEV model. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022130
##### References:
 [1] A. Bensoussan, C. S. Chi, S. C. P. Yam and H. L. Yang, A class of non-zero-sum stochastic differential investment and reinsurance games, Automatica, 50 (2014), 2025-2037.  doi: 10.1016/j.automatica.2014.05.033. [2] A. Bensoussan and J. Frehse, Stochastic games for n players, Journal of Optimization Theory and Applications, 105 (2000), 543-565. doi: 10.1023/A:1004637022496. [3] A. Bensoussan and A. Friedman, Non-zero-sum stochastic differential games with stopping times and free boundary problems, Transactions of the American Mathematical Society, 231 (1977), 275-327.  doi: 10.1090/S0002-9947-1977-0453082-7. [4] J. Bi and J. Cai, Optimal investment-reinsurance strategies with state dependent risk aversion and var constraints in correlated markets, Insurance: Mathematics and Economics, 85 (2019), 1-14.  doi: 10.1016/j.insmatheco.2018.11.007. [5] E. P. Briys, Investment portfolio behavior of non-life insurers: A utility analysis, Insurance: Mathematics and Economics, 4 (1985), 93-98.  doi: 10.1016/0167-6687(85)90003-4. [6] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [7] Y. S. Cao and N. Q. Wan, Optimal proportional reinsurance and investment based on hamilton-jacobi-bellman equation, Insurance: Mathematics and Economics, 45 (2009), 157-162.  doi: 10.1016/j.insmatheco.2009.05.006. [8] B. Cox, Notes on Option Pricing $i$: Constant Elasticity of Variance Diffusions, Stanford: Stanford University, 1975. [9] S. P. David and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 110-128.  doi: 10.1080/10920277.2005.10596214. [10] P. M. Demarzo, R. Kaniel and I. Kremer, Relative wealth concerns and financial bubbles, Review of Financial Studies, 21 (2008), 19-50. [11] C. Deng, X. D. Zeng and H. M. Zhu, Non-zero-sum stochastic differential reinsurance and investment games with default risk, European Journal of Operational Research, 264 (2018), 1144-1158.  doi: 10.1016/j.ejor.2017.06.065. [12] D. Dufresne, The integrated square-root process, (2001). [13] R. J. Elliott, The existence of value in stochastic differential games, Scandinavian Actuarial Journal, 14 (1976), 85-94.  doi: 10.1137/0314006. [14] J. W. Gao, Optimal portfolios for DC pension plans under a CEV model, Insurance: Mathematics and Economics, 44 (2009), 479-490.  doi: 10.1016/j.insmatheco.2009.01.005. [15] C. Gollier and S. Wibaut, Portfolio selection by mutual insurance companies and optimal participating insurance policies, Insurance: Mathematics and Economics, 11 (1992), 237-245.  doi: 10.1016/0167-6687(92)90029-B. [16] C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4. [17] Y. Huang, X. Q. Yang and J. M. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, Journal of Computational and Applied Mathematics, 296 (2015), 443-461.  doi: 10.1016/j.cam.2015.09.032. [18] C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004. [19] Z. Jin, Z. Q. Xu and B. Zou, A perturbation approach to optimal investment, liability ratio, and dividend strategies, Scandinavian Actuarial Journal, 2022 (2021), 1-24.  doi: 10.1080/03461238.2021.1938199. [20] Y. Kahane, Generation of investable funds and the portfolio behavior of the non-life insurers, Journal of Risk Insurance, 45 (1978), 65.  doi: 10.2307/251808. [21] Y. Kahane and D. Nye, A portfolio approach to the property liability insurance industry, Journal of Risk and Insurance, 42 (1975), 579-598.  doi: 10.2307/252154. [22] D. P. Li, Y. Zeng and H. Yang, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scandinavian Actuarial Journal, 2018 (2018), 145-171.  doi: 10.1080/03461238.2017.1309679. [23] Z. Liang and E. Bayraktar, Optimal reinsurance and investment with unobservable claim size and intensity, Insurance: Mathematics and Economics, 55 (2014), 156-166.  doi: 10.1016/j.insmatheco.2014.01.011. [24] J. Z. Liu and K. F. C. Yiu, Optimal stochastic differential games with var constraints, Discrete and Continuous Dynamical Systems, 18 (2013), 1889.  doi: 10.3934/dcdsb.2013.18.1889. [25] X. Peng, F. Chen and W. Wang, Robust optimal investment and reinsurance for an insurer with inside information, Insurance: Mathematics and Economics, 96 (2021), 15-30.  doi: 10.1016/j.insmatheco.2020.10.004. [26] H. Said and H. Mohammed, The multi-player non-zero-sum dynkin game in continuous time, Mathematical Methods of Operations Research, 52 (2014). doi: 10.1137/110855132. [27] Z. Y. Sun, X. X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053. [28] N. Wang, N. Zhang, Z. Jin and L. Y. Qian, Robust non-zero-sum investment and reinsurance game with default risk, Insurance: Mathematics and Economics, 84 (2019), 115-132.  doi: 10.1016/j.insmatheco.2018.09.009. [29] Y. J. Wang, X. M. Rong and H. Zhao, 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, 328 (2017), 414-431.  doi: 10.1016/j.cam.2017.08.001. [30] Z. Wang, J. Xia and L. Zhang, Optimal investment for an insurer: The martingale approach, Insurance: Mathematics and Economics, 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003. [31] B. L. Webb, Investment income in insurance ratemaking, Journal of Insurance Regulation, 1 (1982), 46-76. [32] H. Yang and L. Zhang, Optimal investment for insurer with jump diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [33] X. D. Zeng, A stochastic differential reinsurance game, Journal of Applied Probability, 47 (2010), 335-349.  doi: 10.1239/jap/1276784895.

show all references

##### References:
 [1] A. Bensoussan, C. S. Chi, S. C. P. Yam and H. L. Yang, A class of non-zero-sum stochastic differential investment and reinsurance games, Automatica, 50 (2014), 2025-2037.  doi: 10.1016/j.automatica.2014.05.033. [2] A. Bensoussan and J. Frehse, Stochastic games for n players, Journal of Optimization Theory and Applications, 105 (2000), 543-565. doi: 10.1023/A:1004637022496. [3] A. Bensoussan and A. Friedman, Non-zero-sum stochastic differential games with stopping times and free boundary problems, Transactions of the American Mathematical Society, 231 (1977), 275-327.  doi: 10.1090/S0002-9947-1977-0453082-7. [4] J. Bi and J. Cai, Optimal investment-reinsurance strategies with state dependent risk aversion and var constraints in correlated markets, Insurance: Mathematics and Economics, 85 (2019), 1-14.  doi: 10.1016/j.insmatheco.2018.11.007. [5] E. P. Briys, Investment portfolio behavior of non-life insurers: A utility analysis, Insurance: Mathematics and Economics, 4 (1985), 93-98.  doi: 10.1016/0167-6687(85)90003-4. [6] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [7] Y. S. Cao and N. Q. Wan, Optimal proportional reinsurance and investment based on hamilton-jacobi-bellman equation, Insurance: Mathematics and Economics, 45 (2009), 157-162.  doi: 10.1016/j.insmatheco.2009.05.006. [8] B. Cox, Notes on Option Pricing $i$: Constant Elasticity of Variance Diffusions, Stanford: Stanford University, 1975. [9] S. P. David and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 110-128.  doi: 10.1080/10920277.2005.10596214. [10] P. M. Demarzo, R. Kaniel and I. Kremer, Relative wealth concerns and financial bubbles, Review of Financial Studies, 21 (2008), 19-50. [11] C. Deng, X. D. Zeng and H. M. Zhu, Non-zero-sum stochastic differential reinsurance and investment games with default risk, European Journal of Operational Research, 264 (2018), 1144-1158.  doi: 10.1016/j.ejor.2017.06.065. [12] D. Dufresne, The integrated square-root process, (2001). [13] R. J. Elliott, The existence of value in stochastic differential games, Scandinavian Actuarial Journal, 14 (1976), 85-94.  doi: 10.1137/0314006. [14] J. W. Gao, Optimal portfolios for DC pension plans under a CEV model, Insurance: Mathematics and Economics, 44 (2009), 479-490.  doi: 10.1016/j.insmatheco.2009.01.005. [15] C. Gollier and S. Wibaut, Portfolio selection by mutual insurance companies and optimal participating insurance policies, Insurance: Mathematics and Economics, 11 (1992), 237-245.  doi: 10.1016/0167-6687(92)90029-B. [16] C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4. [17] Y. Huang, X. Q. Yang and J. M. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, Journal of Computational and Applied Mathematics, 296 (2015), 443-461.  doi: 10.1016/j.cam.2015.09.032. [18] C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004. [19] Z. Jin, Z. Q. Xu and B. Zou, A perturbation approach to optimal investment, liability ratio, and dividend strategies, Scandinavian Actuarial Journal, 2022 (2021), 1-24.  doi: 10.1080/03461238.2021.1938199. [20] Y. Kahane, Generation of investable funds and the portfolio behavior of the non-life insurers, Journal of Risk Insurance, 45 (1978), 65.  doi: 10.2307/251808. [21] Y. Kahane and D. Nye, A portfolio approach to the property liability insurance industry, Journal of Risk and Insurance, 42 (1975), 579-598.  doi: 10.2307/252154. [22] D. P. Li, Y. Zeng and H. Yang, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scandinavian Actuarial Journal, 2018 (2018), 145-171.  doi: 10.1080/03461238.2017.1309679. [23] Z. Liang and E. Bayraktar, Optimal reinsurance and investment with unobservable claim size and intensity, Insurance: Mathematics and Economics, 55 (2014), 156-166.  doi: 10.1016/j.insmatheco.2014.01.011. [24] J. Z. Liu and K. F. C. Yiu, Optimal stochastic differential games with var constraints, Discrete and Continuous Dynamical Systems, 18 (2013), 1889.  doi: 10.3934/dcdsb.2013.18.1889. [25] X. Peng, F. Chen and W. Wang, Robust optimal investment and reinsurance for an insurer with inside information, Insurance: Mathematics and Economics, 96 (2021), 15-30.  doi: 10.1016/j.insmatheco.2020.10.004. [26] H. Said and H. Mohammed, The multi-player non-zero-sum dynkin game in continuous time, Mathematical Methods of Operations Research, 52 (2014). doi: 10.1137/110855132. [27] Z. Y. Sun, X. X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053. [28] N. Wang, N. Zhang, Z. Jin and L. Y. Qian, Robust non-zero-sum investment and reinsurance game with default risk, Insurance: Mathematics and Economics, 84 (2019), 115-132.  doi: 10.1016/j.insmatheco.2018.09.009. [29] Y. J. Wang, X. M. Rong and H. Zhao, 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, 328 (2017), 414-431.  doi: 10.1016/j.cam.2017.08.001. [30] Z. Wang, J. Xia and L. Zhang, Optimal investment for an insurer: The martingale approach, Insurance: Mathematics and Economics, 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003. [31] B. L. Webb, Investment income in insurance ratemaking, Journal of Insurance Regulation, 1 (1982), 46-76. [32] H. Yang and L. Zhang, Optimal investment for insurer with jump diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [33] X. D. Zeng, A stochastic differential reinsurance game, Journal of Applied Probability, 47 (2010), 335-349.  doi: 10.1239/jap/1276784895.
(a) Effects of $\eta_{1}$ on $q_{1}^*(t)$ in Case 1. (b) Effects of $\eta_{1}$ on $q_{1}^*(t)$ in Case 2. (c) Effects of $\eta_{1}$ on $q_{1}^*(t)$ in Case 3. (d) Effects of $\eta_{1}$ on $q_{1}^*(t)$ in Case 4
(a) Effects of $\beta_{1}$ on $q_{1}^*(t)$ in Case 1. (b) Effects of $\beta_{1}$ on $q_{1}^*(t)$ in Case 2. (c) Effects of $\beta_{1}$ on $q_{1}^*(t)$ in Case 3. (d) Effects of $\beta_{1}$ on $q_{1}^*(t)$ in Case 4
(a) Effects of $K_{1}$ on $q_{1}^*(t)$ in Case 1. (b) Effects of $K_{1}$ on $q_{1}^*(t)$ in Case 2. (c) Effects of $K_{1}$ on $q_{1}^*(t)$ in Case 3. (d) Effects of $K_{1}$ on $q_{1}^*(t)$ in Case 4
(a) Effects of $\beta_{1}$ on $\pi_{1}^*(t)$ in Case 1. (b) Effects of $\beta_{1}$ on $\pi_{1}^*(t)$ in Case 2. (c) Effects of $\beta_{1}$ on $\pi_{1}^*(t)$ in Case 3. (d) Effects of $\beta_{1}$ on $\pi_{1}^*(t)$ in Case 4
(a) Effects of $K_{1}$ on $\pi_{1}^*(t)$ in Case 1. (b) Effects of $K_{1}$ on $\pi_{1}^*(t)$ in Case 2. (c) Effects of $K_{1}$ on $\pi_{1}^*(t)$ in Case 3. (d) Effects of $K_{1}$ on $\pi_{1}^*(t)$ in Case 4
Effects of $\rho$ on $\pi_{1}^*(t)$
Effects of $\rho$ on $q_{1}^*(t)$
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