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July  2021, 17(4): 1887-1912. doi: 10.3934/jimo.2020051

Optimal mean-variance reinsurance in a financial market with stochastic rate of return

1. 

College of Science, Civil Aviation University of China, Tianjin 300300, China

2. 

School of Mathematical Sciences, Nankai University, Tianjin 300071, China

3. 

School of Statistics, Qufu Normal University, Qufu, Shandong 273165, China

* Corresponding author: Zhongyang Sun

Received  June 2019 Revised  December 2019 Published  July 2021 Early access  March 2020

In this paper, we investigate the optimal investment and reinsurance strategies for a mean-variance insurer when the surplus process is represented by a Cramér-Lundberg model. It is assumed that the instantaneous rate of investment return is stochastic and follows an Ornstein-Uhlenbeck (OU) process, which could describe the features of bull and bear markets. To solve the mean-variance optimization problem, we adopt a backward stochastic differential equation (BSDE) approach and derive explicit expressions for both the efficient strategy and efficient frontier. Finally, numerical examples are presented to illustrate our results.

Citation: Yingxu Tian, Junyi Guo, Zhongyang Sun. 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
References:
[1]

P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér Lundberg model, Mathematical Finance, 15 (2005), 261-308.  doi: 10.1111/j.0960-1627.2005.00220.x.

[2]

S. AsmussenB. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.  doi: 10.1007/s007800050075.

[3]

N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1.

[4]

L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constraint risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.

[5]

C. Bender and M. Kohlmann, BSDEs with Stochastic Lipschitz Condition, Universität Konstanz, Fakultät für Mathematik and Informatik, 2000.

[6]

J. N. Bi and J. Y. Guo, Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, Journal of Optimization Theory and Applications, 157 (2013), 252-275.  doi: 10.1007/s10957-012-0138-y.

[7]

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

[8]

T. R. BieleckiH. Q. JinS. R. Pliskaz and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x.

[9]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponentional utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.

[10]

F.J. Fabozzi and J. C. Francis, Mutual fund systematic risk for bull and bear markets: An empirical examination,, Journal of Finance, 34 (1979), 1243-1250.  doi: 10.1111/j.1540-6261.1979.tb00069.x.

[11]

W. H. Fleming and H. M. Soner, Controled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006.

[12]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 28 (2000), 558-602.  doi: 10.1214/aop/1019160253.

[13]

X. LiX. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.

[14]

Z. B. LiangK. C. Yuen and J. Y. Guo, Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance: Mathematics and Economics, 49 (2011), 207-215.  doi: 10.1016/j.insmatheco.2011.04.005.

[15]

A. E. B. Lim, Mean-variance hedging when there are jumps, SIAM Journal on Control and Optimization, 44 (2005), 1893-1922.  doi: 10.1137/040610933.

[16]

A. E. B. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29 (2004), 132-161.  doi: 10.1287/moor.1030.0065.

[17]

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

[18]

R. C. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.

[19]

E. Pardoux and A. Rǎşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied probability, 69. Springer, Switzerland, 2014. doi: 10.1007/978-3-319-05714-9.

[20]

R. Rishel, Optimal portfolio management with partial observation and power utility function, Stochastic Analysis, Control, Optimization and Applications, Systems Control Found. Appl., Birkhäuser Boston, Boston, MA, (1999), 605–619.

[21]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.

[22]

Y. Shen, Mean-variance portfolio selection in a complete market with unbounded random coefficients, Automatica J. IFAC, 55 (2015), 165-175.  doi: 10.1016/j.automatica.2015.03.009.

[23]

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

[24]

Y. ShenX. Zhang and T. K. Siu, Mean-variance portfolio selection under a constant elasticity of variance model, Operations Research Letters, 42 (2014), 337-342.  doi: 10.1016/j.orl.2014.05.008.

[25]

Z. Y. Sun, Upper bounds for ruin probabilities under model uncertainty, Communications in Statistics-Theory and Methods, 48 (2019), 4511-4527.  doi: 10.1080/03610926.2018.1491991.

[26]

Z. Y. Sun and J. Y. Guo, Optimal mean-variance investment and reinsurance problem for an insurer with stochastic volatility, Mathematical Methods of Operations Research, 88 (2018), 59-79.  doi: 10.1007/s00186-017-0628-7.

[27]

Z. Y. Sun and X. P. Guo, Equilibrium for a time-inconsistent stochastic linear-quadratic control system with jumps and its application to the mean-variance problem, Journal of Optimization Theory and Applications, 181 (2019), 383-410.  doi: 10.1007/s10957-018-01471-x.

[28]

Z. Y. Sun, K. C. Yuen and J. Y. Guo, A BSDE approach to a class of dependent risk model of mean-variance insurers with stochastic volatility and no-short selling, Journal of Computational and Applied Mathematics, 366 (2019), 112413, 21 pp. doi: 10.1016/j.cam.2019.112413.

[29]

Z. Y. Sun, X. Zhang and K. C. Yuen, Mean-variance asset-liability management with affine diffusion factor process and a reinsurance option, Scandinavian Actuarial Journal, (2019), DOI: https://doi.org/10.1080/03461238.2019.1658619. doi: 10.1080/03461238.2019.1658619.

[30]

Z. Y. SunX. 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.

[31]

Z. W. WangJ. M. Xia and L. H. 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.

[32]

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

[33]

J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamilton Systems and HJB Equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[34]

M. Zhang and P. Chen, Mean-variance asset-liability management under constant elasticity of variance process, Insurance: Mathematics and Economics, 70 (2016), 11-18.  doi: 10.1016/j.insmatheco.2016.05.019.

[35]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.

[36]

X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.

show all references

References:
[1]

P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér Lundberg model, Mathematical Finance, 15 (2005), 261-308.  doi: 10.1111/j.0960-1627.2005.00220.x.

[2]

S. AsmussenB. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.  doi: 10.1007/s007800050075.

[3]

N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1.

[4]

L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constraint risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.

[5]

C. Bender and M. Kohlmann, BSDEs with Stochastic Lipschitz Condition, Universität Konstanz, Fakultät für Mathematik and Informatik, 2000.

[6]

J. N. Bi and J. Y. Guo, Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, Journal of Optimization Theory and Applications, 157 (2013), 252-275.  doi: 10.1007/s10957-012-0138-y.

[7]

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

[8]

T. R. BieleckiH. Q. JinS. R. Pliskaz and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x.

[9]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponentional utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.

[10]

F.J. Fabozzi and J. C. Francis, Mutual fund systematic risk for bull and bear markets: An empirical examination,, Journal of Finance, 34 (1979), 1243-1250.  doi: 10.1111/j.1540-6261.1979.tb00069.x.

[11]

W. H. Fleming and H. M. Soner, Controled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006.

[12]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 28 (2000), 558-602.  doi: 10.1214/aop/1019160253.

[13]

X. LiX. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.

[14]

Z. B. LiangK. C. Yuen and J. Y. Guo, Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance: Mathematics and Economics, 49 (2011), 207-215.  doi: 10.1016/j.insmatheco.2011.04.005.

[15]

A. E. B. Lim, Mean-variance hedging when there are jumps, SIAM Journal on Control and Optimization, 44 (2005), 1893-1922.  doi: 10.1137/040610933.

[16]

A. E. B. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29 (2004), 132-161.  doi: 10.1287/moor.1030.0065.

[17]

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

[18]

R. C. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.

[19]

E. Pardoux and A. Rǎşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied probability, 69. Springer, Switzerland, 2014. doi: 10.1007/978-3-319-05714-9.

[20]

R. Rishel, Optimal portfolio management with partial observation and power utility function, Stochastic Analysis, Control, Optimization and Applications, Systems Control Found. Appl., Birkhäuser Boston, Boston, MA, (1999), 605–619.

[21]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.

[22]

Y. Shen, Mean-variance portfolio selection in a complete market with unbounded random coefficients, Automatica J. IFAC, 55 (2015), 165-175.  doi: 10.1016/j.automatica.2015.03.009.

[23]

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

[24]

Y. ShenX. Zhang and T. K. Siu, Mean-variance portfolio selection under a constant elasticity of variance model, Operations Research Letters, 42 (2014), 337-342.  doi: 10.1016/j.orl.2014.05.008.

[25]

Z. Y. Sun, Upper bounds for ruin probabilities under model uncertainty, Communications in Statistics-Theory and Methods, 48 (2019), 4511-4527.  doi: 10.1080/03610926.2018.1491991.

[26]

Z. Y. Sun and J. Y. Guo, Optimal mean-variance investment and reinsurance problem for an insurer with stochastic volatility, Mathematical Methods of Operations Research, 88 (2018), 59-79.  doi: 10.1007/s00186-017-0628-7.

[27]

Z. Y. Sun and X. P. Guo, Equilibrium for a time-inconsistent stochastic linear-quadratic control system with jumps and its application to the mean-variance problem, Journal of Optimization Theory and Applications, 181 (2019), 383-410.  doi: 10.1007/s10957-018-01471-x.

[28]

Z. Y. Sun, K. C. Yuen and J. Y. Guo, A BSDE approach to a class of dependent risk model of mean-variance insurers with stochastic volatility and no-short selling, Journal of Computational and Applied Mathematics, 366 (2019), 112413, 21 pp. doi: 10.1016/j.cam.2019.112413.

[29]

Z. Y. Sun, X. Zhang and K. C. Yuen, Mean-variance asset-liability management with affine diffusion factor process and a reinsurance option, Scandinavian Actuarial Journal, (2019), DOI: https://doi.org/10.1080/03461238.2019.1658619. doi: 10.1080/03461238.2019.1658619.

[30]

Z. Y. SunX. 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.

[31]

Z. W. WangJ. M. Xia and L. H. 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.

[32]

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

[33]

J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamilton Systems and HJB Equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[34]

M. Zhang and P. Chen, Mean-variance asset-liability management under constant elasticity of variance process, Insurance: Mathematics and Economics, 70 (2016), 11-18.  doi: 10.1016/j.insmatheco.2016.05.019.

[35]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.

[36]

X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.

Figure 1.  the path of OU process $ m(t) $ with $ \alpha = -0.04 $ and $ \beta = 0.03 $
Figure 2.  the optimal investment strategy with $ \alpha = -0.04, \beta = 0.03 $ and $ \alpha = \beta = 0 $
Figure 3.  the optimal reinsurance strategy with $ \alpha = -0.04, \beta = 0.03 $ and $ \alpha = \beta = 0 $
Figure 4.  the effect of $ \alpha $ on the efficient frontier
Figure 5.  the effect of $ \lambda $ on the efficient frontier
Figure 6.  the effect of $ \eta $ on the efficient frontier
Figure 7.  the effect of $ r $ on the efficient frontier
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