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  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 & Management Optimization, 2021, 17 (2) : 909-936. doi: 10.3934/jimo.2020004
References:
[1]

C. AY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. BensoussanC. C. SiuS. 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.  Google Scholar

[7]

T. BjörkA. 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.  Google Scholar

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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.  Google Scholar

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G. CallegaroM. GaïgiS. 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.  Google Scholar

[10]

R. CarmonaF. DelarueG. 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.  Google Scholar

[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.  Google Scholar

[12]

S. ChenZ. 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.  Google Scholar

[13]

S. ChenH. 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.  Google Scholar

[14]

C. DengX. 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.  Google Scholar

[15]

I. Elsanosi and B. Larssen, Optimal consumption under partial observations for a stochastic system with delay, Preprint Series in Pure Mathematics. Google Scholar

[16]

I. ElsanosiB. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics and Stochastic Reports, 71 (2000), 69-89.  doi: 10.1080/17442500008834259.  Google Scholar

[17]

G. Espinosa and N. Touzi, Optimal investment under relative performance concerns, Mathematical Finance, 25 (2015), 221-257.  doi: 10.1111/mafi.12034.  Google Scholar

[18]

J. FouqueA. 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.  Google Scholar

[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.  Google Scholar

[20]

J. Grandell, A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977 (1977), 37-52.  doi: 10.1080/03461238.1977.10405071.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. HuS. 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.  Google Scholar

[24]

Y. HuangX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. LiW. 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.  Google Scholar

[28]

Z. LiY. 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.  Google Scholar

[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.  Google Scholar

[30]

X. LinC. 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.  Google Scholar

[31]

H. MengS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

L. XuR. 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.  Google Scholar

[37]

M. YanF. 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.  Google Scholar

[38]

X. YangZ. Liang and C. Zhang, Optimal mean-variance reinsurance with delay and multiple classes of dependent risks, Scientia Sinica Mathematica, 47 (2017), 723-756.   Google Scholar

[39]

X. Zeng, A stochastic differential reinsurance game, Journal of Applied Probability, 47 (2010), 335-349.  doi: 10.1239/jap/1276784895.  Google Scholar

[40]

Y. ZengD. 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.  Google Scholar

[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.  Google Scholar

[42]

Z. ZhouT. RenH. 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.  Google Scholar

show all references

References:
[1]

C. AY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. BensoussanC. C. SiuS. 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.  Google Scholar

[7]

T. BjörkA. 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.  Google Scholar

[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.  Google Scholar

[9]

G. CallegaroM. GaïgiS. 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.  Google Scholar

[10]

R. CarmonaF. DelarueG. 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.  Google Scholar

[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.  Google Scholar

[12]

S. ChenZ. 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.  Google Scholar

[13]

S. ChenH. 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.  Google Scholar

[14]

C. DengX. 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.  Google Scholar

[15]

I. Elsanosi and B. Larssen, Optimal consumption under partial observations for a stochastic system with delay, Preprint Series in Pure Mathematics. Google Scholar

[16]

I. ElsanosiB. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics and Stochastic Reports, 71 (2000), 69-89.  doi: 10.1080/17442500008834259.  Google Scholar

[17]

G. Espinosa and N. Touzi, Optimal investment under relative performance concerns, Mathematical Finance, 25 (2015), 221-257.  doi: 10.1111/mafi.12034.  Google Scholar

[18]

J. FouqueA. 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.  Google Scholar

[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.  Google Scholar

[20]

J. Grandell, A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977 (1977), 37-52.  doi: 10.1080/03461238.1977.10405071.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. HuS. 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.  Google Scholar

[24]

Y. HuangX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. LiW. 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.  Google Scholar

[28]

Z. LiY. 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.  Google Scholar

[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.  Google Scholar

[30]

X. LinC. 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.  Google Scholar

[31]

H. MengS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

L. XuR. 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.  Google Scholar

[37]

M. YanF. 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.  Google Scholar

[38]

X. YangZ. Liang and C. Zhang, Optimal mean-variance reinsurance with delay and multiple classes of dependent risks, Scientia Sinica Mathematica, 47 (2017), 723-756.   Google Scholar

[39]

X. Zeng, A stochastic differential reinsurance game, Journal of Applied Probability, 47 (2010), 335-349.  doi: 10.1239/jap/1276784895.  Google Scholar

[40]

Y. ZengD. 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.  Google Scholar

[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.  Google Scholar

[42]

Z. ZhouT. RenH. 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.  Google Scholar

Figure 1.  Optimal reinsurance strategy
Figure 2.  Optimal investment strategy
Figure 3.  Effect of risk aversion parameters on strategies
Figure 4.  Effect of delay parameters on strategies
Table 1.  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 $ $ + $ $ - $ $ + $
Table 2.  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 $ $ + $ $ - $ $ + $
Table 3.  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 $
Table 4.  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 $
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