# American Institute of Mathematical Sciences

March  2021, 17(2): 909-936. doi: 10.3934/jimo.2020004

## A non-zero-sum reinsurance-investment game with delay and asymmetric information

 1 School of Business Administration, Hunan University, Changsha 410082, China 2 School of Business, Hunan Normal University, Changsha 410081, China 3 College of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

* Corresponding author: Yanfei Bai (baiyanfei0709@163.com)

Received  August 2018 Revised  August 2019 Published  March 2021 Early access  January 2020

Fund Project: This research is supported by the National Natural Science Foundation of China (Nos. 71771082, 71801091) and Hunan Provincial Natural Science Foundation of China (No. 2017JJ1012)

In this paper, we investigate a non-zero-sum stochastic differential reinsurance-investment game problem between two insurers. Both insurers can purchase proportional reinsurance and invest in a financial market that contains a risk-free asset and a risky asset. We consider the insurers' wealth processes with delay to characterize the bounded memory feature. For considering the effect of asymmetric information, we assume the insurers have access to different levels of information in the financial market. Each insurer's objective is to maximize the expected utility of its performance relative to its competitor. We derive the Hamilton-Jacobi-Bellman (HJB) equations and the general Nash equilibrium strategies associated with the control problem by applying the dynamic programming principle. For constant absolute risk aversion (CARA) insurers, the explicit Nash equilibrium strategies and the value functions are obtained. Finally, we present some numerical studies to draw economic interpretations and find the following interesting results: (1) the insurer with less information completely ignores its own risk aversion factor, but imitates the investment strategy of its competitor who has more information on the financial market, which is a manifestation of the herd effect in economics; (2) the difference between the effects of different delay weights on the strategies is related to the length of the delay time in the framework of the non-zero-sum stochastic differential game, which illustrates that insurers should rationally estimate the correlation between historical performance and future performance based on their own risk tolerance, especially when decision makers consider historical performance over a long period of time.

Citation: Zhongbao Zhou, Yanfei Bai, Helu Xiao, Xu Chen. 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
##### References:
 [1] C. A, Y. Lai and Y. Shao, Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the cev model, Journal of Computational and Applied Mathematics, 342 (2018), 317-336.  doi: 10.1016/j.cam.2018.03.035. [2] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance: Mathematics and Economics, 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005. [3] C. A and Y. Shao, Portfolio optimization problem with delay under Cox-Ingersoll-Ross model, Journal of Mathematical Finance, 07 (2017), 699-717.  doi: 10.4236/jmf.2017.73037. [4] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [5] 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. [6] A. Bensoussan, C. C. Siu, S. C. P. Yam and H. 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. [7] T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x. [8] 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. [9] G. Callegaro, M. Gaïgi, S. Scotti and C. Sgarra, Optimal investment in markets with over and under-reaction to information, Mathematics and Financial Economics, 11 (2016), 299-322.  doi: 10.1007/s11579-016-0182-8. [10] R. Carmona, F. Delarue, G. Espinosa and N. Touzi, Singular forward-backward stochastic differential equations and emissions derivatives, The Annals of Applied Probability, 23 (2013), 1086-1128.  doi: 10.1214/12-AAP865. [11] L. Chen and H. Yang, Optimal reinsurance and investment strategy with two piece utility function, Journal of Industrial and Management Optimization, 13 (2017), 737-755.  doi: 10.3934/jimo.2016044. [12] S. Chen, Z. Li and K. Li, Optimal investment-reinsurance policy for an insurance company with VaR constraint, Insurance: Mathematics and Economics, 47 (2010), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002. [13] S. Chen, H. Yang and Y. Zeng, Stochastic differential games between two insurers with generalized mean-variance premium principle, ASTIN Bulletin, 48 (2018), 413-434.  doi: 10.1017/asb.2017.35. [14] C. Deng, X. Zeng and H. 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. [15] I. Elsanosi and B. Larssen, Optimal consumption under partial observations for a stochastic system with delay, Preprint Series in Pure Mathematics. [16] I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics and Stochastic Reports, 71 (2000), 69-89.  doi: 10.1080/17442500008834259. [17] G. Espinosa and N. Touzi, Optimal investment under relative performance concerns, Mathematical Finance, 25 (2015), 221-257.  doi: 10.1111/mafi.12034. [18] J. Fouque, A. Papanicolaou and R. Sircar, Perturbation analysis for investment portfolios under partial information with expert opinions, SIAM Journal on Control and Optimization, 55 (2017), 1534-1566.  doi: 10.1137/15M1006854. [19] H. U. Gerber and E. S. W. Shiu, On optimal dividends: From reflection to refraction, Journal of Computational and Applied Mathematics, 186 (2006), 4-22.  doi: 10.1016/j.cam.2005.03.062. [20] J. Grandell, A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977 (1977), 37-52.  doi: 10.1080/03461238.1977.10405071. [21] G. Guan and Z. Liang, A stochastic Nash equilibrium portfolio game between two DC pension funds, Insurance: Mathematics and Economics, 70 (2016), 237-244.  doi: 10.1016/j.insmatheco.2016.06.015. [22] S. L. Hansen, Optimal consumption and investment strategies with partial and private information in a multi-asset setting, Mathematics and Financial Economics, 7 (2012), 305-340.  doi: 10.1007/s11579-012-0086-1. [23] D. Hu, S. Chen and H. Wang, Robust reinsurance contracts with uncertainty about jump risk, European Journal of Operational Research, 266 (2018), 1175-1188.  doi: 10.1016/j.ejor.2017.10.061. [24] Y. Huang, X. Yang and J. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, Journal of Computational and Applied Mathematics, 296 (2016), 443-461.  doi: 10.1016/j.cam.2015.09.032. [25] R. Korn and P. Wilmott, Optimal portfolios under the threat of a crash, International Journal of Theoretical and Applied Finance, 5 (2002), 171-187.  doi: 10.1142/S0219024902001407. [26] P. Lakner, Utility maximization with partial information, Stochastic Processes and Their Applications, 56 (1995), 247-273.  doi: 10.1016/0304-4149(94)00073-3. [27] P. Li, W. Zhao and W. Zhou, Ruin probabilities and optimal investment when the stock price follows an exponential Lévy process, Applied Mathematics and Computation, 259 (2015), 1030-1045.  doi: 10.1016/j.amc.2014.12.042. [28] Z. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002. [29] X. Lin and Y. Qian, Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model, Scandinavian Actuarial Journal, 2016 (2016), 646-671.  doi: 10.1080/03461238.2015.1048710. [30] X. Lin, C. Zhang and T. K. Siu, Stochastic differential portfolio games for an insurer in a jump-diffusion risk process, Mathematical Methods of Operations Research, 75 (2012), 83-100.  doi: 10.1007/s00186-011-0376-z. [31] H. Meng, S. Li and Z. Jin, A reinsurance game between two insurance companies with nonlinear risk processes, Insurance: Mathematics and Economics, 62 (2015), 91-97.  doi: 10.1016/j.insmatheco.2015.03.008. [32] R. Merton, On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 8 (1980), 323-361.  doi: 10.3386/w0444. [33] C. S. Pun and H. Y. Wong, Robust non-zero-sum stochastic differential reinsurance game, Insurance: Mathematics and Economics, 68 (2016), 169-177.  doi: 10.1016/j.insmatheco.2016.02.007. [34] W. Putschögl and J. Sass, Optimal consumption and investment under partial information, Decisions in Economics and Finance, 31 (2008), 137-170.  doi: 10.1007/s10203-008-0082-3. [35] Y. Shen and Y. Zeng, Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach, Insurance: Mathematics and Economics, 57 (2014), 1-12.  doi: 10.1016/j.insmatheco.2014.04.004. [36] L. Xu, R. Wang and D. Yao, On maximizing the expected terminal utility by investment and reinsurance, Journal of Industrial and Management Optimization, 4 (2008), 801-815.  doi: 10.3934/jimo.2008.4.801. [37] M. Yan, F. Peng and S. Zhang, A reinsurance and investment game between two insurance companies with the different opinions about some extra information, Insurance: Mathematics and Economics, 75 (2017), 58-70.  doi: 10.1016/j.insmatheco.2017.04.002. [38] X. Yang, Z. Liang and C. Zhang, Optimal mean-variance reinsurance with delay and multiple classes of dependent risks, Scientia Sinica Mathematica, 47 (2017), 723-756. [39] X. Zeng, A stochastic differential reinsurance game, Journal of Applied Probability, 47 (2010), 335-349.  doi: 10.1239/jap/1276784895. [40] Y. Zeng, D. Li and A. Gu, Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance: Mathematics and Economics, 66 (2016), 138-152.  doi: 10.1016/j.insmatheco.2015.10.012. [41] H. Zhao and X. Rong, On the constant elasticity of variance model for the utility maximization problem with multiple risky assets, IMA Journal of Management Mathematics, 28 (2017), 299-320.  doi: 10.1093/imaman/dpv011. [42] Z. Zhou, T. Ren, H. Xiao and W. Liu, Time-consistent investment and reinsurance strategies for insurers under multi-period mean-variance formulation with generalized correlated returns, Journal of Management Science and Engineering, 4 (2019), 142-157.  doi: 10.1016/j.jmse.2019.05.003.

show all references

##### References:
 [1] C. A, Y. Lai and Y. Shao, Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the cev model, Journal of Computational and Applied Mathematics, 342 (2018), 317-336.  doi: 10.1016/j.cam.2018.03.035. [2] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance: Mathematics and Economics, 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005. [3] C. A and Y. Shao, Portfolio optimization problem with delay under Cox-Ingersoll-Ross model, Journal of Mathematical Finance, 07 (2017), 699-717.  doi: 10.4236/jmf.2017.73037. [4] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [5] 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. [6] A. Bensoussan, C. C. Siu, S. C. P. Yam and H. 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. [7] T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x. [8] 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. [9] G. Callegaro, M. Gaïgi, S. Scotti and C. Sgarra, Optimal investment in markets with over and under-reaction to information, Mathematics and Financial Economics, 11 (2016), 299-322.  doi: 10.1007/s11579-016-0182-8. [10] R. Carmona, F. Delarue, G. Espinosa and N. Touzi, Singular forward-backward stochastic differential equations and emissions derivatives, The Annals of Applied Probability, 23 (2013), 1086-1128.  doi: 10.1214/12-AAP865. [11] L. Chen and H. Yang, Optimal reinsurance and investment strategy with two piece utility function, Journal of Industrial and Management Optimization, 13 (2017), 737-755.  doi: 10.3934/jimo.2016044. [12] S. Chen, Z. Li and K. Li, Optimal investment-reinsurance policy for an insurance company with VaR constraint, Insurance: Mathematics and Economics, 47 (2010), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002. [13] S. Chen, H. Yang and Y. Zeng, Stochastic differential games between two insurers with generalized mean-variance premium principle, ASTIN Bulletin, 48 (2018), 413-434.  doi: 10.1017/asb.2017.35. [14] C. Deng, X. Zeng and H. 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. [15] I. Elsanosi and B. Larssen, Optimal consumption under partial observations for a stochastic system with delay, Preprint Series in Pure Mathematics. [16] I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics and Stochastic Reports, 71 (2000), 69-89.  doi: 10.1080/17442500008834259. [17] G. Espinosa and N. Touzi, Optimal investment under relative performance concerns, Mathematical Finance, 25 (2015), 221-257.  doi: 10.1111/mafi.12034. [18] J. Fouque, A. Papanicolaou and R. Sircar, Perturbation analysis for investment portfolios under partial information with expert opinions, SIAM Journal on Control and Optimization, 55 (2017), 1534-1566.  doi: 10.1137/15M1006854. [19] H. U. Gerber and E. S. W. Shiu, On optimal dividends: From reflection to refraction, Journal of Computational and Applied Mathematics, 186 (2006), 4-22.  doi: 10.1016/j.cam.2005.03.062. [20] J. Grandell, A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977 (1977), 37-52.  doi: 10.1080/03461238.1977.10405071. [21] G. Guan and Z. Liang, A stochastic Nash equilibrium portfolio game between two DC pension funds, Insurance: Mathematics and Economics, 70 (2016), 237-244.  doi: 10.1016/j.insmatheco.2016.06.015. [22] S. L. Hansen, Optimal consumption and investment strategies with partial and private information in a multi-asset setting, Mathematics and Financial Economics, 7 (2012), 305-340.  doi: 10.1007/s11579-012-0086-1. [23] D. Hu, S. Chen and H. Wang, Robust reinsurance contracts with uncertainty about jump risk, European Journal of Operational Research, 266 (2018), 1175-1188.  doi: 10.1016/j.ejor.2017.10.061. [24] Y. Huang, X. Yang and J. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, Journal of Computational and Applied Mathematics, 296 (2016), 443-461.  doi: 10.1016/j.cam.2015.09.032. [25] R. Korn and P. Wilmott, Optimal portfolios under the threat of a crash, International Journal of Theoretical and Applied Finance, 5 (2002), 171-187.  doi: 10.1142/S0219024902001407. [26] P. Lakner, Utility maximization with partial information, Stochastic Processes and Their Applications, 56 (1995), 247-273.  doi: 10.1016/0304-4149(94)00073-3. [27] P. Li, W. Zhao and W. Zhou, Ruin probabilities and optimal investment when the stock price follows an exponential Lévy process, Applied Mathematics and Computation, 259 (2015), 1030-1045.  doi: 10.1016/j.amc.2014.12.042. [28] Z. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002. [29] X. Lin and Y. Qian, Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model, Scandinavian Actuarial Journal, 2016 (2016), 646-671.  doi: 10.1080/03461238.2015.1048710. [30] X. Lin, C. Zhang and T. K. Siu, Stochastic differential portfolio games for an insurer in a jump-diffusion risk process, Mathematical Methods of Operations Research, 75 (2012), 83-100.  doi: 10.1007/s00186-011-0376-z. [31] H. Meng, S. Li and Z. Jin, A reinsurance game between two insurance companies with nonlinear risk processes, Insurance: Mathematics and Economics, 62 (2015), 91-97.  doi: 10.1016/j.insmatheco.2015.03.008. [32] R. Merton, On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 8 (1980), 323-361.  doi: 10.3386/w0444. [33] C. S. Pun and H. Y. Wong, Robust non-zero-sum stochastic differential reinsurance game, Insurance: Mathematics and Economics, 68 (2016), 169-177.  doi: 10.1016/j.insmatheco.2016.02.007. [34] W. Putschögl and J. Sass, Optimal consumption and investment under partial information, Decisions in Economics and Finance, 31 (2008), 137-170.  doi: 10.1007/s10203-008-0082-3. [35] Y. Shen and Y. Zeng, Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach, Insurance: Mathematics and Economics, 57 (2014), 1-12.  doi: 10.1016/j.insmatheco.2014.04.004. [36] L. Xu, R. Wang and D. Yao, On maximizing the expected terminal utility by investment and reinsurance, Journal of Industrial and Management Optimization, 4 (2008), 801-815.  doi: 10.3934/jimo.2008.4.801. [37] M. Yan, F. Peng and S. Zhang, A reinsurance and investment game between two insurance companies with the different opinions about some extra information, Insurance: Mathematics and Economics, 75 (2017), 58-70.  doi: 10.1016/j.insmatheco.2017.04.002. [38] X. Yang, Z. Liang and C. Zhang, Optimal mean-variance reinsurance with delay and multiple classes of dependent risks, Scientia Sinica Mathematica, 47 (2017), 723-756. [39] X. Zeng, A stochastic differential reinsurance game, Journal of Applied Probability, 47 (2010), 335-349.  doi: 10.1239/jap/1276784895. [40] Y. Zeng, D. Li and A. Gu, Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance: Mathematics and Economics, 66 (2016), 138-152.  doi: 10.1016/j.insmatheco.2015.10.012. [41] H. Zhao and X. Rong, On the constant elasticity of variance model for the utility maximization problem with multiple risky assets, IMA Journal of Management Mathematics, 28 (2017), 299-320.  doi: 10.1093/imaman/dpv011. [42] Z. Zhou, T. Ren, H. Xiao and W. Liu, Time-consistent investment and reinsurance strategies for insurers under multi-period mean-variance formulation with generalized correlated returns, Journal of Management Science and Engineering, 4 (2019), 142-157.  doi: 10.1016/j.jmse.2019.05.003.
Optimal reinsurance strategy
Optimal investment strategy
Effect of risk aversion parameters on strategies
Effect of delay parameters on strategies
The properties of $q_i^{\ast}(t)$
 $\frac{\partial q_i^{\ast}(t)}{\partial h_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial\eta_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial\bar{\eta}_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial \alpha_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial \gamma_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial k_i}$ $\bar{H}_i<1$ $\bar{H}_i = 1$ $\bar{H}_i>1$ $+$ $+$ $+$ $-$ $0$ $+$ $-$ $+$
 $\frac{\partial q_i^{\ast}(t)}{\partial h_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial\eta_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial\bar{\eta}_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial \alpha_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial \gamma_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial k_i}$ $\bar{H}_i<1$ $\bar{H}_i = 1$ $\bar{H}_i>1$ $+$ $+$ $+$ $-$ $0$ $+$ $-$ $+$
The properties of $l_i^{\ast}(t)$
 $\frac{\partial l_i^{\ast}(t)}{\partial h_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial\eta_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial\bar{\eta}_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial \alpha_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial \gamma_1}$ $\frac{\partial l_i^{\ast}(t)}{\partial k_i}$ $\bar{H}_i<1$ $\bar{H}_i = 1$ $\bar{H}_i>1$ $+$ $+$ $+$ $-$ $0$ $+$ $-$ $+$
 $\frac{\partial l_i^{\ast}(t)}{\partial h_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial\eta_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial\bar{\eta}_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial \alpha_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial \gamma_1}$ $\frac{\partial l_i^{\ast}(t)}{\partial k_i}$ $\bar{H}_i<1$ $\bar{H}_i = 1$ $\bar{H}_i>1$ $+$ $+$ $+$ $-$ $0$ $+$ $-$ $+$
The parameter values of the financial market
 $r_0$ $r$ $\sigma$ $\beta$ $s_0$ $T$ $0.05$ $0.1$ $0.4$ $1$ $1$ $10$
 $r_0$ $r$ $\sigma$ $\beta$ $s_0$ $T$ $0.05$ $0.1$ $0.4$ $1$ $1$ $10$
The parameter values of insurers
 Insurer $1$ Insurer $2$ Parameter Value Parameter Value $\mu_1$ $5$ $\mu_2$ $1$ $\sigma_1$ $8$ $\sigma_2$ $5$ $h_1$ $2$ $h_2$ $3$ $\alpha_1$ $0.5$ $\alpha_2$ $0.3$ $\eta_1$ $0.05$ $\eta_2$ $/$ $\gamma_1$ $0.3$ $\gamma_2$ $0.1$ $k_1$ $0.4$ $k_2$ $0.3$ $\rho$ $0.5$ $\rho$ $0.5$
 Insurer $1$ Insurer $2$ Parameter Value Parameter Value $\mu_1$ $5$ $\mu_2$ $1$ $\sigma_1$ $8$ $\sigma_2$ $5$ $h_1$ $2$ $h_2$ $3$ $\alpha_1$ $0.5$ $\alpha_2$ $0.3$ $\eta_1$ $0.05$ $\eta_2$ $/$ $\gamma_1$ $0.3$ $\gamma_2$ $0.1$ $k_1$ $0.4$ $k_2$ $0.3$ $\rho$ $0.5$ $\rho$ $0.5$
 [1] Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control and Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013 [2] Shuaiqi Zhang, Jie Xiong, Xin Zhang. Optimal investment problem with delay under partial information. Mathematical Control and Related Fields, 2020, 10 (2) : 365-378. doi: 10.3934/mcrf.2020001 [3] Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001 [4] Chloé Jimenez. A zero sum differential game with correlated informations on the initial position. A case with a continuum of initial positions. Journal of Dynamics and Games, 2021, 8 (3) : 233-266. doi: 10.3934/jdg.2021009 [5] Abd El-Monem A. Megahed, Ebrahim A. Youness, Hebatallah K. Arafat. Optimization method in counter terrorism: Min-Max zero-sum differential game approach. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022013 [6] Apostolis Pavlou. Asymmetric information in a bilateral monopoly. Journal of Dynamics and Games, 2016, 3 (2) : 169-189. doi: 10.3934/jdg.2016009 [7] Sheng Li, Wei Yuan, Peimin Chen. Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022068 [8] Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080 [9] Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control and Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028 [10] Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial and Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27 [11] Lin Xu, Rongming Wang, Dingjun Yao. On maximizing the expected terminal utility by investment and reinsurance. Journal of Industrial and Management Optimization, 2008, 4 (4) : 801-815. doi: 10.3934/jimo.2008.4.801 [12] Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics and Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020 [13] Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zero-sum games. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3901-3931. doi: 10.3934/dcds.2015.35.3901 [14] Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial and Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008 [15] Yan Zhang, Peibiao Zhao. Optimal reinsurance-investment problem with dependent risks based on Legendre transform. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1457-1479. doi: 10.3934/jimo.2019011 [16] Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial and Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044 [17] Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009 [18] Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control and Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003 [19] Nan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Markowitz's mean-variance optimization with investment and constrained reinsurance. Journal of Industrial and Management Optimization, 2017, 13 (1) : 375-397. doi: 10.3934/jimo.2016022 [20] Xiaoyu Xing, Caixia Geng. Optimal investment-reinsurance strategy in the correlated insurance and financial markets. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021120

2021 Impact Factor: 1.411