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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. Asmussen, B. 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. Bi, Z. 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. Bielecki, H. Q. Jin, S. 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. Li, X. 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. Liang, K. 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. Shen, X. 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. 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. [31] Z. W. Wang, J. 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. Asmussen, B. 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. Bi, Z. 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. Bielecki, H. Q. Jin, S. 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. Li, X. 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. Liang, K. 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. Shen, X. 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. 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. [31] Z. W. Wang, J. 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.
the path of OU process $m(t)$ with $\alpha = -0.04$ and $\beta = 0.03$
the optimal investment strategy with $\alpha = -0.04, \beta = 0.03$ and $\alpha = \beta = 0$
the optimal reinsurance strategy with $\alpha = -0.04, \beta = 0.03$ and $\alpha = \beta = 0$
the effect of $\alpha$ on the efficient frontier
the effect of $\lambda$ on the efficient frontier
the effect of $\eta$ on the efficient frontier
the effect of $r$ on the efficient frontier
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