
-
Previous Article
Back-ordered inventory model with inflation in a cloudy-fuzzy environment
- JIMO Home
- This Issue
-
Next Article
Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |






[1] |
Yan Zhang, Peibiao Zhao, Xinghu Teng, Lei Mao. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2139-2159. doi: 10.3934/jimo.2020062 |
[2] |
Sumon Sarkar, Bibhas C. Giri. Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021048 |
[3] |
Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021049 |
[4] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[5] |
Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021025 |
[6] |
Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 |
[7] |
Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209 |
[8] |
Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021025 |
[9] |
Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021020 |
[10] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[11] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[12] |
Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021026 |
[13] |
Jan Rychtář, Dewey T. Taylor. Moran process and Wright-Fisher process favor low variability. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3491-3504. doi: 10.3934/dcdsb.2020242 |
[14] |
Andrés Contreras, Juan Peypouquet. Forward-backward approximation of nonlinear semigroups in finite and infinite horizon. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021051 |
[15] |
Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021072 |
[16] |
Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021008 |
[17] |
Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021023 |
[18] |
Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021107 |
[19] |
Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021012 |
[20] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]