
-
Previous Article
A variational inequality approach for constrained multifacility Weber problem under gauge
- JIMO Home
- This Issue
-
Next Article
Second-order optimality conditions for cone constrained multi-objective optimization
Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle
1. | School of Finance, Nanjing University of Finance and Economics, Nanjing 210023, China |
2. | School of Statistics, East China Normal University, Shanghai 200241, China |
This paper assumes that an insurer can control the dividend, refinancing and reinsurance strategies dynamically. Particularly, the reinsurance is provided by two reinsurers and the variance premium principle is applied in pricing insurance contracts. Using the optimal control method, we identify the optimal strategies for maximizing the insurance company's value. Meanwhile, the effects of transaction costs and terminal value at bankruptcy are investigated. The results turn out that the insurer should consider refinancing when and only when the transaction costs and terminal value are relatively low. Also, it should buy less reinsurance when the surplus increases, while the proportion of risk allocation between two reinsurers remains constant. When the dividend rate is unbounded, dividends should be paid according to the barrier strategy. When the dividend rate is restricted, dividends should be distributed according to the threshold strategy. Some examples are provided to illustrate the implementation of our results.
References:
[1] |
S. Asmussen and M. Taksar,
Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.
doi: 10.1016/S0167-6687(96)00017-0. |
[2] |
L. Bai, J. Guo and H. Zhang,
Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes, Quantitative Finance, 10 (2010), 1163-1172.
doi: 10.1080/14697680902968005. |
[3] |
A. Barth and S. Moreno-Bromberg,
Optimal risk and liquidity management with costly refinancing opportunities, Insurance: Mathematics and Economics, 57 (2014), 31-45.
doi: 10.1016/j.insmatheco.2014.05.001. |
[4] |
A. Cadenillas, T. Choulli, M. Taksar and L. Zhang,
Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm, Mathematical Finance, 16 (2006), 181-202.
doi: 10.1111/j.1467-9965.2006.00267.x. |
[5] |
M. Chen, X. Peng and J. Guo,
Optimal dividend problem with a nonlinear regular-singular stochastic control, Insurance: Mathematics and Economics, 52 (2013), 448-456.
doi: 10.1016/j.insmatheco.2013.02.010. |
[6] |
M. Chen and K. C. Yuen,
Optimal dividend and reinsurance in the presence of two reinsurers, Journal of Applied Probability, 53 (2016), 554-571.
doi: 10.1017/jpr.2016.20. |
[7] | B. De Finetti, Su un'impostzione alternativa della teoria collettiva del rischio, in Transactions of the XVth International Congress of Actuaries, Congrès Internationald'Actuaires, New York, 1957. Google Scholar |
[8] |
W. Fleming and H. Soner,
Controlled Markov Process and Viscosity Solutions, Springer-Verlag, 1993. |
[9] |
H. U. Gerber and E. S. W. Shiu,
On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93.
doi: 10.1080/10920277.2006.10596249. |
[10] |
J. Grandell,
Aspects of Risk Theory, Springer-Verlag, 1991.
doi: 10.1007/978-1-4613-9058-9. |
[11] |
H. Guan and Z. Liang,
Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs, Insurance: Mathematics and Economics, 54 (2014), 109-122.
doi: 10.1016/j.insmatheco.2013.11.003. |
[12] |
L. He and Z. Liang,
Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: Mathematics and Economics, 44 (2009), 88-94.
doi: 10.1016/j.insmatheco.2008.10.001. |
[13] |
B. H$\phi $gaard and M. Taksar,
Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.
doi: 10.1111/1467-9965.00066. |
[14] |
Z. Liang and V. Young,
Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.
doi: 10.1016/j.insmatheco.2012.02.005. |
[15] |
W. Liu and Y. Hu,
Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy, Statistics and Probability Letters, 84 (2014), 121-130.
doi: 10.1016/j.spl.2013.09.034. |
[16] |
R. L. Loeffen,
An optimal dividends problem with transaction costs for spectrally negative L$\acute{e}$vy processes, Insurance: Mathematics and Economics, 45 (2009), 41-48.
doi: 10.1016/j.insmatheco.2009.03.002. |
[17] |
A. L$φ$kka and M. Zervos,
Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961.
doi: 10.1016/j.insmatheco.2007.10.013. |
[18] |
H. Meng, Optimal impulse control with variance premium principle, Science China Mathematics (in Chinese), 43 (2013), 925-939. Google Scholar |
[19] |
H. Meng and T. K. Siu,
On optimal reinsurance, dividend and reinvestment strategies, Economic Modelling, 28 (2011), 211-218.
doi: 10.1016/j.econmod.2010.09.009. |
[20] |
X. Peng, M. Chen and J. Guo,
Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.
doi: 10.1016/j.insmatheco.2012.08.004. |
[21] |
M. Taksar,
Dependence of the optimal risk control decisions on the terminal value for a financial corporation, Annals of Operations Research, 98 (2000), 89-99.
doi: 10.1023/A:1019239920624. |
[22] |
J. Xu and M. Zhou,
Optimal risk control and dividend distribution policies for a diffusion model with terminal value, Mathematical and Computer Modelling, 56 (2012), 180-190.
doi: 10.1016/j.mcm.2011.12.041. |
[23] |
D. Yao, H. Yang and R. Wang,
Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.
doi: 10.1016/j.econmod.2013.10.026. |
[24] |
M. Zhou and K. C. Yuen,
Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.
doi: 10.1016/j.econmod.2011.09.007. |
show all references
References:
[1] |
S. Asmussen and M. Taksar,
Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.
doi: 10.1016/S0167-6687(96)00017-0. |
[2] |
L. Bai, J. Guo and H. Zhang,
Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes, Quantitative Finance, 10 (2010), 1163-1172.
doi: 10.1080/14697680902968005. |
[3] |
A. Barth and S. Moreno-Bromberg,
Optimal risk and liquidity management with costly refinancing opportunities, Insurance: Mathematics and Economics, 57 (2014), 31-45.
doi: 10.1016/j.insmatheco.2014.05.001. |
[4] |
A. Cadenillas, T. Choulli, M. Taksar and L. Zhang,
Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm, Mathematical Finance, 16 (2006), 181-202.
doi: 10.1111/j.1467-9965.2006.00267.x. |
[5] |
M. Chen, X. Peng and J. Guo,
Optimal dividend problem with a nonlinear regular-singular stochastic control, Insurance: Mathematics and Economics, 52 (2013), 448-456.
doi: 10.1016/j.insmatheco.2013.02.010. |
[6] |
M. Chen and K. C. Yuen,
Optimal dividend and reinsurance in the presence of two reinsurers, Journal of Applied Probability, 53 (2016), 554-571.
doi: 10.1017/jpr.2016.20. |
[7] | B. De Finetti, Su un'impostzione alternativa della teoria collettiva del rischio, in Transactions of the XVth International Congress of Actuaries, Congrès Internationald'Actuaires, New York, 1957. Google Scholar |
[8] |
W. Fleming and H. Soner,
Controlled Markov Process and Viscosity Solutions, Springer-Verlag, 1993. |
[9] |
H. U. Gerber and E. S. W. Shiu,
On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93.
doi: 10.1080/10920277.2006.10596249. |
[10] |
J. Grandell,
Aspects of Risk Theory, Springer-Verlag, 1991.
doi: 10.1007/978-1-4613-9058-9. |
[11] |
H. Guan and Z. Liang,
Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs, Insurance: Mathematics and Economics, 54 (2014), 109-122.
doi: 10.1016/j.insmatheco.2013.11.003. |
[12] |
L. He and Z. Liang,
Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: Mathematics and Economics, 44 (2009), 88-94.
doi: 10.1016/j.insmatheco.2008.10.001. |
[13] |
B. H$\phi $gaard and M. Taksar,
Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.
doi: 10.1111/1467-9965.00066. |
[14] |
Z. Liang and V. Young,
Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.
doi: 10.1016/j.insmatheco.2012.02.005. |
[15] |
W. Liu and Y. Hu,
Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy, Statistics and Probability Letters, 84 (2014), 121-130.
doi: 10.1016/j.spl.2013.09.034. |
[16] |
R. L. Loeffen,
An optimal dividends problem with transaction costs for spectrally negative L$\acute{e}$vy processes, Insurance: Mathematics and Economics, 45 (2009), 41-48.
doi: 10.1016/j.insmatheco.2009.03.002. |
[17] |
A. L$φ$kka and M. Zervos,
Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961.
doi: 10.1016/j.insmatheco.2007.10.013. |
[18] |
H. Meng, Optimal impulse control with variance premium principle, Science China Mathematics (in Chinese), 43 (2013), 925-939. Google Scholar |
[19] |
H. Meng and T. K. Siu,
On optimal reinsurance, dividend and reinvestment strategies, Economic Modelling, 28 (2011), 211-218.
doi: 10.1016/j.econmod.2010.09.009. |
[20] |
X. Peng, M. Chen and J. Guo,
Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.
doi: 10.1016/j.insmatheco.2012.08.004. |
[21] |
M. Taksar,
Dependence of the optimal risk control decisions on the terminal value for a financial corporation, Annals of Operations Research, 98 (2000), 89-99.
doi: 10.1023/A:1019239920624. |
[22] |
J. Xu and M. Zhou,
Optimal risk control and dividend distribution policies for a diffusion model with terminal value, Mathematical and Computer Modelling, 56 (2012), 180-190.
doi: 10.1016/j.mcm.2011.12.041. |
[23] |
D. Yao, H. Yang and R. Wang,
Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.
doi: 10.1016/j.econmod.2013.10.026. |
[24] |
M. Zhou and K. C. Yuen,
Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.
doi: 10.1016/j.econmod.2011.09.007. |



Re | ||||||||
0.6409 | 3.2729 | - | - | - | 1.2438 | 0.0184 | n | |
0.6116 | 3.4762 | - | - | - | 1.4845 | 0.1096 | n | |
0.5892 | 3.6368 | 0.6035 | 3.5331 | 0.8915 | 1.7414 | 0.2610 | y | |
0.5719 | 3.7665 | 0.6236 | 3.3919 | 0.9531 | 2.0110 | 0.4598 | y | |
0.5582 | 3.8730 | 0.6455 | 3.2418 | 1.0286 | 2.2903 | 0.6979 | y | |
0.5472 | 3.9621 | 0.6695 | 3.0787 | 1.1255 | 2.5775 | 0.9700 | y |
Re | ||||||||
0.6409 | 3.2729 | - | - | - | 1.2438 | 0.0184 | n | |
0.6116 | 3.4762 | - | - | - | 1.4845 | 0.1096 | n | |
0.5892 | 3.6368 | 0.6035 | 3.5331 | 0.8915 | 1.7414 | 0.2610 | y | |
0.5719 | 3.7665 | 0.6236 | 3.3919 | 0.9531 | 2.0110 | 0.4598 | y | |
0.5582 | 3.8730 | 0.6455 | 3.2418 | 1.0286 | 2.2903 | 0.6979 | y | |
0.5472 | 3.9621 | 0.6695 | 3.0787 | 1.1255 | 2.5775 | 0.9700 | y |
Re | ||||||||
0.5892 | 3.6368 | - | - | - | 1.7414 | 0.3145 | n | |
0.5892 | 3.6368 | - | - | - | 1.7414 | 0.2610 | n | |
0.5892 | - | - | - | 3.6368 | 1.7414 | 0.2146 | n | |
0.5892 | 3.6368 | 0.6160 | 3.4445 | 0.9557 | 1.7414 | 0.1743 | y | |
0.5892 | 3.6368 | 0.6035 | 3.5331 | 0.8915 | 1.7414 | 0.1395 | y | |
0.5892 | 3.6368 | 0.5927 | 3.6116 | 0.8387 | 1.7414 | 0.1095 | y |
Re | ||||||||
0.5892 | 3.6368 | - | - | - | 1.7414 | 0.3145 | n | |
0.5892 | 3.6368 | - | - | - | 1.7414 | 0.2610 | n | |
0.5892 | - | - | - | 3.6368 | 1.7414 | 0.2146 | n | |
0.5892 | 3.6368 | 0.6160 | 3.4445 | 0.9557 | 1.7414 | 0.1743 | y | |
0.5892 | 3.6368 | 0.6035 | 3.5331 | 0.8915 | 1.7414 | 0.1395 | y | |
0.5892 | 3.6368 | 0.5927 | 3.6116 | 0.8387 | 1.7414 | 0.1095 | y |
Re | ||||||||
0.5892 | 3.6368 | 0.6625 | 3.1266 | 0.4850 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6368 | 0.6364 | 3.3034 | 0.6619 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6368 | 0.6180 | 3.4308 | 0.7893 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6368 | 0.6035 | 3.5331 | 0.8915 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6368 | 0.5916 | 3.6195 | 0.9779 | 1.7414 | 0.2610 | y | |
0.5892 | - | - | - | 3.6368 | 1.7414 | 0.2610 | n |
Re | ||||||||
0.5892 | 3.6368 | 0.6625 | 3.1266 | 0.4850 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6368 | 0.6364 | 3.3034 | 0.6619 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6368 | 0.6180 | 3.4308 | 0.7893 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6368 | 0.6035 | 3.5331 | 0.8915 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6368 | 0.5916 | 3.6195 | 0.9779 | 1.7414 | 0.2610 | y | |
0.5892 | - | - | - | 3.6368 | 1.7414 | 0.2610 | n |
Re | ||||||||
0.4368 | 5.5029 | 0.6035 | 3.5331 | 0.8915 | 9.6961 | 6.2083 | y | |
0.4449 | 5.2679 | 0.6035 | 3.5331 | 0.8915 | 7.4376 | 4.4668 | y | |
0.4600 | 4.9497 | 0.6035 | 3.5331 | 0.8915 | 5.2780 | 2.8168 | y | |
0.4940 | 4.4728 | 0.6035 | 3.5331 | 0.8915 | 3.3073 | 1.3414 | y | |
0.5892 | 3.6368 | 0.6035 | 3.5331 | 0.8915 | 1.7414 | 0.2610 | y | |
0.8209 | 2.0027 | - | - | - | 0.9351 | - | n |
Re | ||||||||
0.4368 | 5.5029 | 0.6035 | 3.5331 | 0.8915 | 9.6961 | 6.2083 | y | |
0.4449 | 5.2679 | 0.6035 | 3.5331 | 0.8915 | 7.4376 | 4.4668 | y | |
0.4600 | 4.9497 | 0.6035 | 3.5331 | 0.8915 | 5.2780 | 2.8168 | y | |
0.4940 | 4.4728 | 0.6035 | 3.5331 | 0.8915 | 3.3073 | 1.3414 | y | |
0.5892 | 3.6368 | 0.6035 | 3.5331 | 0.8915 | 1.7414 | 0.2610 | y | |
0.8209 | 2.0027 | - | - | - | 0.9351 | - | n |
Re | ||||||||
0.6036 | 2.3715 | - | - | - | 1.5666 | 0.1530 | n | |
0.5918 | 3.0719 | 0.6011 | 3.0056 | 0.8841 | 1.7066 | 0.2380 | y | |
0.5893 | 3.5344 | 0.6035 | 3.4318 | 0.8913 | 1.7403 | 0.2602 | y | |
0.5892 | 3.5862 | 0.6035 | 3.4827 | 0.8915 | 1.7411 | 0.2608 | y | |
0.5892 | 3.6318 | 0.6035 | 3.5280 | 0.8915 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6363 | 0.6035 | 3.5325 | 0.8915 | 1.7414 | 0.2610 | y |
Re | ||||||||
0.6036 | 2.3715 | - | - | - | 1.5666 | 0.1530 | n | |
0.5918 | 3.0719 | 0.6011 | 3.0056 | 0.8841 | 1.7066 | 0.2380 | y | |
0.5893 | 3.5344 | 0.6035 | 3.4318 | 0.8913 | 1.7403 | 0.2602 | y | |
0.5892 | 3.5862 | 0.6035 | 3.4827 | 0.8915 | 1.7411 | 0.2608 | y | |
0.5892 | 3.6318 | 0.6035 | 3.5280 | 0.8915 | 1.7414 | 0.2610 | y | |
0.5892 | 3.6363 | 0.6035 | 3.5325 | 0.8915 | 1.7414 | 0.2610 | y |
[1] |
Gongpin Cheng, Lin Xu. Optimal size of business and dividend strategy in a nonlinear model with refinancing and liquidation value. Mathematical Control & Related Fields, 2017, 7 (1) : 1-19. doi: 10.3934/mcrf.2017001 |
[2] |
Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080 |
[3] |
Lin Xu, Rongming Wang, Dingjun Yao. On maximizing the expected terminal utility by investment and reinsurance. Journal of Industrial & Management Optimization, 2008, 4 (4) : 801-815. doi: 10.3934/jimo.2008.4.801 |
[4] |
Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009 |
[5] |
Yingxu Tian, Junyi Guo, Zhongyang Sun. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1887-1912. doi: 10.3934/jimo.2020051 |
[6] |
Nan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Markowitz's mean-variance optimization with investment and constrained reinsurance. Journal of Industrial & Management Optimization, 2017, 13 (1) : 375-397. doi: 10.3934/jimo.2016022 |
[7] |
Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067 |
[8] |
Gongpin Cheng, Rongming Wang, Dingjun Yao. Optimal dividend and capital injection strategy with excess-of-loss reinsurance and transaction costs. Journal of Industrial & Management Optimization, 2018, 14 (1) : 371-395. doi: 10.3934/jimo.2017051 |
[9] |
Tomás Caraballo, Tran Bao Ngoc, Tran Ngoc Thach, Nguyen Huy Tuan. On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4299-4323. doi: 10.3934/dcdsb.2020289 |
[10] |
Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174 |
[11] |
Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial & Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044 |
[12] |
Ming Yan, Hongtao Yang, Lei Zhang, Shuhua Zhang. Optimal investment-reinsurance policy with regime switching and value-at-risk constraint. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2195-2211. doi: 10.3934/jimo.2019050 |
[13] |
Arjuna Flenner, Gary A. Hewer, Charles S. Kenney. Two dimensional histogram analysis using the Helmholtz principle. Inverse Problems & Imaging, 2008, 2 (4) : 485-525. doi: 10.3934/ipi.2008.2.485 |
[14] |
K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624 |
[15] |
Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021 |
[16] |
Zhimin Zhang, Yang Yang, Chaolin Liu. On a perturbed compound Poisson model with varying premium rates. Journal of Industrial & Management Optimization, 2017, 13 (2) : 721-736. doi: 10.3934/jimo.2016043 |
[17] |
Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759 |
[18] |
Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834 |
[19] |
M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072 |
[20] |
Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]