doi: 10.3934/jimo.2021140
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Open-loop equilibrium mean-variance reinsurance, new business and investment strategies with constraints

1. 

School of Statistics, East China Normal University, Shanghai 200062, China

2. 

Key Laboratory of Advanced Theory and, Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai 200241, China

* Corresponding author: jqwei@stat.ecnu.edu.cn

Received  March 2021 Revised  May 2021 Early access August 2021

Fund Project: This work was supported by the Central Universities (2020ECNU-HWFW003), the 111 Project (B14019); the National Natural Science Foundation of China (12071147, 12071146, 11571113, 11601157, 11601320); the China Scholarship Council (201906140044); Graduate Research Innovation Program of School of Statistics, East China Normal University

In this paper, we study a general mean-variance reinsurance, new business and investment problem, where the claim processes of original and new businesses are modeled by two different risk processes and the safety loadings of reinsurance and new business are different. The retention level of the insurer is constrained in $ [0,1] $ and the controls of new business and risky investment are required to be non-negative. This model relaxes the limitations of those in existing research. By using the projection onto the convex set controls valued in, we obtain an open-loop equilibrium reinsurance-new business-investment strategy explicitly. We also show that the obtained equilibrium strategy is the optimal one among all deterministic strategies in the sense that it yields the smallest mean-variance cost. In the case where original and new businesses are the same, the equilibrium strategy is given in closed-form and its sensitivities to safety loadings are shown by numerical examples. At last, by comparing with the case where acquiring new business is prohibited, we show that allowing writing new policies indeed improves the performance of the insurer's risk management.

Citation: Liming Zhang, Rongming Wang, Jiaqin Wei. Open-loop equilibrium mean-variance reinsurance, new business and investment strategies with constraints. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021140
References:
[1]

S. AsmussenB. J. Christensen and M. Taksar, Portfolio size as function of the premium: Modelling and optimization, Stochastics, 85 (2010), 575-588.  doi: 10.1080/17442508.2013.797426.  Google Scholar

[2]

L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[3]

L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.  Google Scholar

[4]

N. Bäuerle, Benchmark and mean-variance problems for insurers, Math. Methods Oper. Res., 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1.  Google Scholar

[5]

J. Bi and J. Cai, Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets, Insurance Math. Econom., 85 (2019), 1-14.  doi: 10.1016/j.insmatheco.2018.11.007.  Google Scholar

[6]

J. BiZ. Liang and F. Xu, Optimal mean–variance investment and reinsurance problems for the risk model with common shock dependence, Insurance Math. Econom., 70 (2016), 245-258.  doi: 10.1016/j.insmatheco.2016.06.012.  Google Scholar

[7]

T. BjörkM. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.  Google Scholar

[8]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, Preprint SSRN, 55 (2010). Google Scholar

[9]

T. BjörkA. Murgoci and X. Zhou, Mean–variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.  Google Scholar

[10]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar

[11]

L. ChenL. QianY. Shen and W. Wang, Constrained investment–reinsurance optimization with regime switching under variance premium principle, Insurance Math. Econom., 71 (2016), 253-267.  doi: 10.1016/j.insmatheco.2016.09.009.  Google Scholar

[12]

P. Chen and S. C. P. Yam, Optimal proportional reinsurance and investment with regime-switching for mean-variance insurers, Insurance Math. Econom., 53 (2013), 871-883.  doi: 10.1016/j.insmatheco.2013.10.004.  Google Scholar

[13]

S. ChenZ. Li and K. Li, Optimal investment-reinsurance policy for an insurance company with VaR constraint, Insurance Math. Econom., 47 (2013), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002.  Google Scholar

[14]

I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Math. Financ. Econ., 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6.  Google Scholar

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[16]

B. Han and H. Y. Wong, Backward stochastic differential equations in finance, Available at SSRN 3182387, (2019). Google Scholar

[17]

C. Hipp and M. Taksar, Backward stochastic differential equations in finance, Math. Finance, 26 (2000), 185-192.   Google Scholar

[18]

J. B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer Science & Business Media, 2012. doi: 10.1007/978-3-642-56468-0.  Google Scholar

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B. Højgaard, Optimal dynamic premium control in non-life insurance. Maximizing dividend pay-outs, Scand. Actuar. J., 4 (2002), 225-245.  doi: 10.1080/03461230110106291.  Google Scholar

[20]

Y. HuH. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.  doi: 10.1137/110853960.  Google Scholar

[21]

D. LiX. Rong and H. Zhao, Time-consistent reinsurance–investment strategy for an insurer and a reinsurer with mean–variance criterion under the CEV model, J. Comput. Appl. Math., 283 (2015), 142-162.  doi: 10.1016/j.cam.2015.01.038.  Google Scholar

[22]

Y. Li and Z. Li, Optimal time-consistent investment and reinsurance strategies for mean–variance insurers with state dependent risk aversion, Insurance Math. Econom., 53 (2013), 86-97.  doi: 10.1016/j.insmatheco.2013.03.008.  Google Scholar

[23]

Z. LiY. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.  Google Scholar

[24]

X. Liang and V. R. Young, Minimizing the probability of ruin: Optimal per-loss reinsurance, Insurance Math. Econom., 82 (2018), 181-190.  doi: 10.1016/j.insmatheco.2018.07.005.  Google Scholar

[25]

Z. Liang and E. Bayraktar, Optimal reinsurance and investment with unobservable claim size and intensity, Insurance Math. Econom., 55 (2014), 156-166.  doi: 10.1016/j.insmatheco.2014.01.011.  Google Scholar

[26]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.   Google Scholar

[27]

F. Menoncin and E. Vigna, Mean-variance target-based optimisation for defined contribution pension schemes in a stochastic framework, Insurance Math. Econom., 76 (2017), 172-184.  doi: 10.1016/j.insmatheco.2017.08.002.  Google Scholar

[28]

D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 110-128.  doi: 10.1080/10920277.2005.10596214.  Google Scholar

[29]

Y. Shen and Y. Zeng, Optimal investment–reinsurance strategy for mean–variance insurers with square-root factor process, Insurance Math. Econom., 62 (2015), 118-137.  doi: 10.1016/j.insmatheco.2015.03.009.  Google Scholar

[30]

R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, The Review of Economic Studies, 23 (1955), 128-143.  doi: 10.1007/978-1-349-15492-0_10.  Google Scholar

[31]

H. WangR. Wang and J. Wei, Time-consistent investment-proportional reinsurance strategy with random coefficients for mean–variance insurers, Insurance Math. Econom., 85 (2019), 104-114.  doi: 10.1016/j.insmatheco.2019.01.002.  Google Scholar

[32]

T. Wang and J. Wei, Time-consistent mean–variance asset–liability management with random coefficients, Insurance Math. Econom., 77 (2017), 84-96.  doi: 10.1016/j.insmatheco.2017.08.011.  Google Scholar

[33]

T. Yan and H. Y. Wong, Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility, Insurance Math. Econom., 90 (2020), 105-119.  doi: 10.1016/j.insmatheco.2019.11.003.  Google Scholar

[34]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[35]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations-time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523.  doi: 10.1090/tran/6502.  Google Scholar

[36]

Y. Zeng and Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001.  Google Scholar

[37]

Y. ZengZ. Li and Y. Lai, Time-consistent investment and reinsurance strategies for mean–variance insurers with jumps, Insurance Math. Econom., 52 (2013), 498-507.  doi: 10.1016/j.insmatheco.2013.02.007.  Google Scholar

[38]

L. ZhangR. Wang and J. Wei, Optimal mean-variance reinsurance and investment strategy with constraints in a non-Markovian regime-switching model, Stat. Theory Relat. Fields, 4 (2020), 214-227.  doi: 10.1080/24754269.2020.1719356.  Google Scholar

[39]

X. ZhangH. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean–variance premium principle and no-short selling, Insurance Math. Econom., 67 (2016), 125-132.  doi: 10.1016/j.insmatheco.2016.01.001.  Google Scholar

[40]

H. ZhaoY. Shen and Y. Zeng, Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security, J. Math. Anal. Appl., 437 (2016), 1036-1057.  doi: 10.1016/j.jmaa.2016.01.035.  Google Scholar

show all references

References:
[1]

S. AsmussenB. J. Christensen and M. Taksar, Portfolio size as function of the premium: Modelling and optimization, Stochastics, 85 (2010), 575-588.  doi: 10.1080/17442508.2013.797426.  Google Scholar

[2]

L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[3]

L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.  Google Scholar

[4]

N. Bäuerle, Benchmark and mean-variance problems for insurers, Math. Methods Oper. Res., 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1.  Google Scholar

[5]

J. Bi and J. Cai, Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets, Insurance Math. Econom., 85 (2019), 1-14.  doi: 10.1016/j.insmatheco.2018.11.007.  Google Scholar

[6]

J. BiZ. Liang and F. Xu, Optimal mean–variance investment and reinsurance problems for the risk model with common shock dependence, Insurance Math. Econom., 70 (2016), 245-258.  doi: 10.1016/j.insmatheco.2016.06.012.  Google Scholar

[7]

T. BjörkM. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.  Google Scholar

[8]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, Preprint SSRN, 55 (2010). Google Scholar

[9]

T. BjörkA. Murgoci and X. Zhou, Mean–variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.  Google Scholar

[10]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar

[11]

L. ChenL. QianY. Shen and W. Wang, Constrained investment–reinsurance optimization with regime switching under variance premium principle, Insurance Math. Econom., 71 (2016), 253-267.  doi: 10.1016/j.insmatheco.2016.09.009.  Google Scholar

[12]

P. Chen and S. C. P. Yam, Optimal proportional reinsurance and investment with regime-switching for mean-variance insurers, Insurance Math. Econom., 53 (2013), 871-883.  doi: 10.1016/j.insmatheco.2013.10.004.  Google Scholar

[13]

S. ChenZ. Li and K. Li, Optimal investment-reinsurance policy for an insurance company with VaR constraint, Insurance Math. Econom., 47 (2013), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002.  Google Scholar

[14]

I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Math. Financ. Econ., 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6.  Google Scholar

[15]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.  Google Scholar

[16]

B. Han and H. Y. Wong, Backward stochastic differential equations in finance, Available at SSRN 3182387, (2019). Google Scholar

[17]

C. Hipp and M. Taksar, Backward stochastic differential equations in finance, Math. Finance, 26 (2000), 185-192.   Google Scholar

[18]

J. B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer Science & Business Media, 2012. doi: 10.1007/978-3-642-56468-0.  Google Scholar

[19]

B. Højgaard, Optimal dynamic premium control in non-life insurance. Maximizing dividend pay-outs, Scand. Actuar. J., 4 (2002), 225-245.  doi: 10.1080/03461230110106291.  Google Scholar

[20]

Y. HuH. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.  doi: 10.1137/110853960.  Google Scholar

[21]

D. LiX. Rong and H. Zhao, Time-consistent reinsurance–investment strategy for an insurer and a reinsurer with mean–variance criterion under the CEV model, J. Comput. Appl. Math., 283 (2015), 142-162.  doi: 10.1016/j.cam.2015.01.038.  Google Scholar

[22]

Y. Li and Z. Li, Optimal time-consistent investment and reinsurance strategies for mean–variance insurers with state dependent risk aversion, Insurance Math. Econom., 53 (2013), 86-97.  doi: 10.1016/j.insmatheco.2013.03.008.  Google Scholar

[23]

Z. LiY. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.  Google Scholar

[24]

X. Liang and V. R. Young, Minimizing the probability of ruin: Optimal per-loss reinsurance, Insurance Math. Econom., 82 (2018), 181-190.  doi: 10.1016/j.insmatheco.2018.07.005.  Google Scholar

[25]

Z. Liang and E. Bayraktar, Optimal reinsurance and investment with unobservable claim size and intensity, Insurance Math. Econom., 55 (2014), 156-166.  doi: 10.1016/j.insmatheco.2014.01.011.  Google Scholar

[26]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.   Google Scholar

[27]

F. Menoncin and E. Vigna, Mean-variance target-based optimisation for defined contribution pension schemes in a stochastic framework, Insurance Math. Econom., 76 (2017), 172-184.  doi: 10.1016/j.insmatheco.2017.08.002.  Google Scholar

[28]

D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 110-128.  doi: 10.1080/10920277.2005.10596214.  Google Scholar

[29]

Y. Shen and Y. Zeng, Optimal investment–reinsurance strategy for mean–variance insurers with square-root factor process, Insurance Math. Econom., 62 (2015), 118-137.  doi: 10.1016/j.insmatheco.2015.03.009.  Google Scholar

[30]

R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, The Review of Economic Studies, 23 (1955), 128-143.  doi: 10.1007/978-1-349-15492-0_10.  Google Scholar

[31]

H. WangR. Wang and J. Wei, Time-consistent investment-proportional reinsurance strategy with random coefficients for mean–variance insurers, Insurance Math. Econom., 85 (2019), 104-114.  doi: 10.1016/j.insmatheco.2019.01.002.  Google Scholar

[32]

T. Wang and J. Wei, Time-consistent mean–variance asset–liability management with random coefficients, Insurance Math. Econom., 77 (2017), 84-96.  doi: 10.1016/j.insmatheco.2017.08.011.  Google Scholar

[33]

T. Yan and H. Y. Wong, Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility, Insurance Math. Econom., 90 (2020), 105-119.  doi: 10.1016/j.insmatheco.2019.11.003.  Google Scholar

[34]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[35]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations-time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523.  doi: 10.1090/tran/6502.  Google Scholar

[36]

Y. Zeng and Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001.  Google Scholar

[37]

Y. ZengZ. Li and Y. Lai, Time-consistent investment and reinsurance strategies for mean–variance insurers with jumps, Insurance Math. Econom., 52 (2013), 498-507.  doi: 10.1016/j.insmatheco.2013.02.007.  Google Scholar

[38]

L. ZhangR. Wang and J. Wei, Optimal mean-variance reinsurance and investment strategy with constraints in a non-Markovian regime-switching model, Stat. Theory Relat. Fields, 4 (2020), 214-227.  doi: 10.1080/24754269.2020.1719356.  Google Scholar

[39]

X. ZhangH. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean–variance premium principle and no-short selling, Insurance Math. Econom., 67 (2016), 125-132.  doi: 10.1016/j.insmatheco.2016.01.001.  Google Scholar

[40]

H. ZhaoY. Shen and Y. Zeng, Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security, J. Math. Anal. Appl., 437 (2016), 1036-1057.  doi: 10.1016/j.jmaa.2016.01.035.  Google Scholar

Figure 1.  Equilibrium strategies, $a_{1} = 0.2, \ b_{1} = 0.4, \ a_{2} = 0.3, \ b_{21} = 0, \ b_{22} = 0.5, \ r = 0.05, \ \mu = 0.2, \ \sigma_{1} = \sigma_{2} = 0, \ \sigma_{3} = 0.3, \ T = 10, \ \lambda = 1 $
Figure 2.  Equilibrium retention level and new business strategy, $a_{1} = a_{2} = 0.2, \ b_{1} = b_{21} = 0.4, \ b_{22} = 0, \ r = 0.05, \ \mu = 0.2, \ \sigma_{2} = 0, \ \sigma_{3} = 0.3, \ T = 10, \ \lambda = 1 $
Figure 3.  Equilibrium investment strategy, $a_{1} = a_{2} = 0.2, \ b_{1} = b_{21} = 0.4, \ b_{22} = 0, \ r = 0.05, \ \mu = 0.2, \ \sigma_{2} = 0, \ \sigma_{3} = 0.3, \ T = 10, \ \lambda = 1$
Table 1.  Relations among $ x_{i} $, $ i = 1,2,3,4 $
Cases Relations
$ \sigma_{1}>\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}} $ $ 0 <x_{1}<x_{3}<x_{2}<x_{4} $
$ \sigma_{1}=\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}} $ $ 0<x_{1}<x_{3}=x_{2}=x_{4} $
$ \sigma_{1}\in\left(0,\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}}\right) $ $ 0<x_{1}<x_{4}<x_{2}<x_{3} $
$ \sigma_=0$ $ 0=x_{1}<x_{2}=x_{4} $
$ \sigma_{1}\in\left(-\frac{\sigma_{3}^{2}b_{1}P_{0}}{\lambda\left(\mu-r\right)},0\right) $ $ x_{3}<x_{1}<0<x_{2}<x_{4} $
$ \sigma_{1}\leq-\frac{\sigma_{3}^{2}b_{1}P_{0}}{\lambda\left(\mu-r\right)} $ $ x_{3}<x_{1}<x_{2}\leq0<x_{4} $
Cases Relations
$ \sigma_{1}>\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}} $ $ 0 <x_{1}<x_{3}<x_{2}<x_{4} $
$ \sigma_{1}=\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}} $ $ 0<x_{1}<x_{3}=x_{2}=x_{4} $
$ \sigma_{1}\in\left(0,\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}}\right) $ $ 0<x_{1}<x_{4}<x_{2}<x_{3} $
$ \sigma_=0$ $ 0=x_{1}<x_{2}=x_{4} $
$ \sigma_{1}\in\left(-\frac{\sigma_{3}^{2}b_{1}P_{0}}{\lambda\left(\mu-r\right)},0\right) $ $ x_{3}<x_{1}<0<x_{2}<x_{4} $
$ \sigma_{1}\leq-\frac{\sigma_{3}^{2}b_{1}P_{0}}{\lambda\left(\mu-r\right)} $ $ x_{3}<x_{1}<x_{2}\leq0<x_{4} $
Table 2.  Comparisons of $ \widehat{\boldsymbol{u}}_{1}$ and $\boldsymbol{u}_{1}^{*}$
Cases $p^{\ast}- \widehat{p}$ $\pi^{\ast}- \widehat{\pi}$
$\eta\leq x_{1}$ 0 $\frac{\lambda a_{1}\sigma_{1}\left(x_{1}-\eta\right)}{\sigma_{3}^{2}b_{1}P_{0}}\geq0$
$\sigma_{1}\geq\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}}$ $\eta\in\left(x_{1}, x_{2}\right)$ 0 0
$\eta\in\left[x_{3}, x_{2}\right)$ $\frac{\lambda a_{1}\sigma_{1}^{2}\left(\eta-x_{3}\right)}{\sigma_{3}^{2}b_{1}^{2}P_{0}}\geq0$ 0
$\eta\in\left[x_{2}, x_{4}\right)$ $\frac{\lambda a_{1}\left(x_{4}-\eta\right)}{b_{1}^{2} P_{0}}>0$ 0
$\eta\geq x_{4}$ 0 0
$\eta\leq x_{1}$ 0 $\frac{\lambda a_{1}\sigma_{1}\left(\sigma_{1}^{2}+\sigma_{3}^{2}\right)\left(x_{1}-\eta\right)}{\left(\sigma_{1}^{2}+\sigma_{3}^{2}\right)\sigma_{3}^{2}b_{1}P_{0}}\geq0$
$\sigma_{1}\in\left(0, \frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}}\right)$ $\eta\in\left(x_{1}, x_{2}\right)$ 0 0
$\eta\in\left[x_{2}, x_{3}\right)$ 0 $\frac{\lambda a_{1}\sigma_{1}\left(x_{2}-\eta\right)}{\sigma_{3}^{2}b_{1}P_{0}}\leq0$
$\eta\geq x_{3}$ 0 $\frac{\sigma_{1}b_{1}P_{0}-\lambda\left(\mu-r\right)}{\left(\sigma_{1}^{2}+\sigma_{3}^{2}\right)P_{0}} <0$
$\sigma_{1}\leq0$ $\eta<x_{2}$ 0 0
$\eta\geq x_{2}$ 0 $\frac{\lambda a_{1}\sigma_{1}\left(x_{2}-\eta\right)}{\sigma_{3}^{2}b_{1}P_{0}}\leq0$
Cases $p^{\ast}- \widehat{p}$ $\pi^{\ast}- \widehat{\pi}$
$\eta\leq x_{1}$ 0 $\frac{\lambda a_{1}\sigma_{1}\left(x_{1}-\eta\right)}{\sigma_{3}^{2}b_{1}P_{0}}\geq0$
$\sigma_{1}\geq\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}}$ $\eta\in\left(x_{1}, x_{2}\right)$ 0 0
$\eta\in\left[x_{3}, x_{2}\right)$ $\frac{\lambda a_{1}\sigma_{1}^{2}\left(\eta-x_{3}\right)}{\sigma_{3}^{2}b_{1}^{2}P_{0}}\geq0$ 0
$\eta\in\left[x_{2}, x_{4}\right)$ $\frac{\lambda a_{1}\left(x_{4}-\eta\right)}{b_{1}^{2} P_{0}}>0$ 0
$\eta\geq x_{4}$ 0 0
$\eta\leq x_{1}$ 0 $\frac{\lambda a_{1}\sigma_{1}\left(\sigma_{1}^{2}+\sigma_{3}^{2}\right)\left(x_{1}-\eta\right)}{\left(\sigma_{1}^{2}+\sigma_{3}^{2}\right)\sigma_{3}^{2}b_{1}P_{0}}\geq0$
$\sigma_{1}\in\left(0, \frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}}\right)$ $\eta\in\left(x_{1}, x_{2}\right)$ 0 0
$\eta\in\left[x_{2}, x_{3}\right)$ 0 $\frac{\lambda a_{1}\sigma_{1}\left(x_{2}-\eta\right)}{\sigma_{3}^{2}b_{1}P_{0}}\leq0$
$\eta\geq x_{3}$ 0 $\frac{\sigma_{1}b_{1}P_{0}-\lambda\left(\mu-r\right)}{\left(\sigma_{1}^{2}+\sigma_{3}^{2}\right)P_{0}} <0$
$\sigma_{1}\leq0$ $\eta<x_{2}$ 0 0
$\eta\geq x_{2}$ 0 $\frac{\lambda a_{1}\sigma_{1}\left(x_{2}-\eta\right)}{\sigma_{3}^{2}b_{1}P_{0}}\leq0$
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