
-
Previous Article
Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison
- JIMO Home
- This Issue
-
Next Article
An efficient low complexity algorithm for box-constrained weighted maximin dispersion problem
Optimal reinsurance-investment and dividends problem with fixed transaction costs
1. | School of Mathematics, Southeast University, Nanjing, Jiangsu Province, 211189, China |
2. | Department of Mathematics, SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, China |
3. | School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China |
In this paper, we consider the dividend optimization problem for a financial corporation with fixed transaction costs. Besides the dividend control, the financial corporation takes proportional reinsurance to reduce risk and invests its reserve in a financial market consisting of a risk-free asset (bond) and a risky asset (stock). Because of the presence of the fixed transaction costs, the problem becomes a mixed classical-impulse stochastic control problem. We solve this problem explicitly and construct the value function together with the optimal policy.
References:
[1] |
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. |
[2] |
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. |
[3] |
S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, Singapore: World Scientific, 2010.
doi: 10.1142/9789814282536. |
[4] |
A. Bensoussan and J. Lions, Nouvelle formulation de problèmes de contrôle impulsionnel et applications, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1189–A1192. |
[5] |
A. Bensoussan and J. Lions, Impulse Control and Quasivariational Inequalities, $\mu $, Gauthier-Villars, Montrouge, 1984, Translated from the French by J. M. Cole. |
[6] |
S. Browne,
Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.
doi: 10.1287/moor.20.4.937. |
[7] |
A. Cadenillas,
Consumption-investment problems with transaction costs: Survey and open problems, Mathematical Methods of Operations Research, 51 (2000), 43-68.
doi: 10.1007/s001860050002. |
[8] |
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. |
[9] |
A. Cadenillas and F. Zapatero,
Classical and impulse stochastic control of the exchange rate using interest rates and reserves, Mathematical Finance, 10 (2000), 141-156.
doi: 10.1111/1467-9965.00086. |
[10] |
T. Choulli, M. Taksar and X. Y. Zhou,
Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quantitative Finance, 1 (2001), 573-596.
doi: 10.1088/1469-7688/1/6/301. |
[11] |
T. Choulli, M. Taksar and X. Zhou,
A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979.
doi: 10.1137/S0363012900382667. |
[12] |
A. Dixit,
A simplified treatment of the theory of optimal regulation of Brownian motion, Journal of Economic Dynamics and Control, 15 (1991), 657-673.
doi: 10.1016/0165-1889(91)90037-2. |
[13] |
B. Dumas,
Super contact and related optimality conditions, Journal of Economic Dynamics and Control, 15 (1991), 675-685.
doi: 10.1016/0165-1889(91)90038-3. |
[14] |
J. Harrison, T. Sellke and A. Taylor,
Impulse control of Brownian motion, Mathematics of Operations Research, 8 (1983), 454-466.
doi: 10.1287/moor.8.3.454. |
[15] |
B. Højgaard 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. |
[16] |
B. Højgaard and M. Taksar,
Optimal risk control for a large corporation in the presence of returns on investments, Finance and Stochastics, 5 (2001), 527-547.
doi: 10.1007/PL00000042. |
[17] |
B. Højgaard and M. Taksar,
Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 4 (2004), 315-327.
doi: 10.1088/1469-7688/4/3/007. |
[18] |
B. Højgaard and M. Taksar,
Optimal proportional reinsurance policies for diffusion models, Scandinavian Actuarial Journal, 1998 (1998), 166-180.
|
[19] |
M. Jeanblanc-Picque and A. Shiryaev,
Optimization of the flow of dividends, Russian Mathematical Surveys, 50 (1995), 257-277.
doi: 10.1070/RM1995v050n02ABEH002054. |
[20] |
R. Korn,
Optimal inpulse control when control actions have random consequences, Mathematics of Operations Research, 22 (1997), 639-667.
doi: 10.1287/moor.22.3.639. |
[21] |
R. Korn,
Portfolio optimisation with strictly positive transaction costs and impulse control, Finance and Stochastics, 2 (1998), 85-114.
doi: 10.1007/s007800050034. |
[22] |
P. Li, M. Zhou and C. Yin,
Optimal reinsurance with both proportional and fixed costs, Statistics & Probability Letters, 106 (2015), 134-141.
doi: 10.1016/j.spl.2015.06.024. |
[23] |
J. Paulsen and H. Gjessing,
Ruin theory with stochastic return on investments, Advances in Applied Probability, 29 (1997), 965-985.
doi: 10.2307/1427849. |
[24] |
S. Peng, Backward stochastic differential equations-stochastic optimization theory and viscosity solutions of hjb equations, Topics on Stochastic Analysis, 85–138. Google Scholar |
[25] |
S. Richard,
Optimal impulse control of a diffusion process with both fixed and proportional costs of control, SIAM J. Control Optim., 15 (1977), 79-91.
doi: 10.1137/0315007. |
[26] |
M. Taksar,
Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.
doi: 10.1007/s001860050001. |
[27] |
Z. Wu and Z. Yu,
Dynamic programming principle for one kind of stochastic recursive optimal control problem and hamilton–jacobi–bellman equation, SIAM Journal on Control and Optimization, 47 (2008), 2616-2641.
doi: 10.1137/060671917. |
[28] |
Z. Wu and L. Zhang,
The corporate optimal portfolio and consumption choice problem in the real project with borrowing rate higher than deposit rate, Applied mathematics and computation, 175 (2006), 1596-1608.
doi: 10.1016/j.amc.2005.09.007. |
[29] |
J. Xiong, S. Zhang, H Zhao and X. Zeng,
Optimal proportional reinsurance and investment problem with jump-diffusion risk process under effect of inside information, Frontiers of Mathematics in China, 9 (2014), 965-982.
doi: 10.1007/s11464-014-0403-5. |
[30] |
H. Yang and L. Zhang,
Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.
doi: 10.1016/j.insmatheco.2005.06.009. |
[31] |
C. Yin and K. C. Yuen,
Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs, Journal of Industrial and Management Optimization, 11 (2015), 1247-1262.
doi: 10.3934/jimo.2015.11.1247. |
[32] |
S. Zhang,
Impulse stochastic control for the optimization of the dividend payments of the compound Poisson risk model perturbed by diffusion, Stochastic Analysis and Applications, 30 (2012), 642-661.
doi: 10.1080/07362994.2012.684324. |
[33] |
X. Zhang, M. Zhou and J. Y. Guo,
Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Applied Stochastic Models in Business and Industry, 23 (2007), 63-71.
doi: 10.1002/asmb.637. |
show all references
References:
[1] |
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. |
[2] |
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. |
[3] |
S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, Singapore: World Scientific, 2010.
doi: 10.1142/9789814282536. |
[4] |
A. Bensoussan and J. Lions, Nouvelle formulation de problèmes de contrôle impulsionnel et applications, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1189–A1192. |
[5] |
A. Bensoussan and J. Lions, Impulse Control and Quasivariational Inequalities, $\mu $, Gauthier-Villars, Montrouge, 1984, Translated from the French by J. M. Cole. |
[6] |
S. Browne,
Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.
doi: 10.1287/moor.20.4.937. |
[7] |
A. Cadenillas,
Consumption-investment problems with transaction costs: Survey and open problems, Mathematical Methods of Operations Research, 51 (2000), 43-68.
doi: 10.1007/s001860050002. |
[8] |
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. |
[9] |
A. Cadenillas and F. Zapatero,
Classical and impulse stochastic control of the exchange rate using interest rates and reserves, Mathematical Finance, 10 (2000), 141-156.
doi: 10.1111/1467-9965.00086. |
[10] |
T. Choulli, M. Taksar and X. Y. Zhou,
Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quantitative Finance, 1 (2001), 573-596.
doi: 10.1088/1469-7688/1/6/301. |
[11] |
T. Choulli, M. Taksar and X. Zhou,
A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979.
doi: 10.1137/S0363012900382667. |
[12] |
A. Dixit,
A simplified treatment of the theory of optimal regulation of Brownian motion, Journal of Economic Dynamics and Control, 15 (1991), 657-673.
doi: 10.1016/0165-1889(91)90037-2. |
[13] |
B. Dumas,
Super contact and related optimality conditions, Journal of Economic Dynamics and Control, 15 (1991), 675-685.
doi: 10.1016/0165-1889(91)90038-3. |
[14] |
J. Harrison, T. Sellke and A. Taylor,
Impulse control of Brownian motion, Mathematics of Operations Research, 8 (1983), 454-466.
doi: 10.1287/moor.8.3.454. |
[15] |
B. Højgaard 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. |
[16] |
B. Højgaard and M. Taksar,
Optimal risk control for a large corporation in the presence of returns on investments, Finance and Stochastics, 5 (2001), 527-547.
doi: 10.1007/PL00000042. |
[17] |
B. Højgaard and M. Taksar,
Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 4 (2004), 315-327.
doi: 10.1088/1469-7688/4/3/007. |
[18] |
B. Højgaard and M. Taksar,
Optimal proportional reinsurance policies for diffusion models, Scandinavian Actuarial Journal, 1998 (1998), 166-180.
|
[19] |
M. Jeanblanc-Picque and A. Shiryaev,
Optimization of the flow of dividends, Russian Mathematical Surveys, 50 (1995), 257-277.
doi: 10.1070/RM1995v050n02ABEH002054. |
[20] |
R. Korn,
Optimal inpulse control when control actions have random consequences, Mathematics of Operations Research, 22 (1997), 639-667.
doi: 10.1287/moor.22.3.639. |
[21] |
R. Korn,
Portfolio optimisation with strictly positive transaction costs and impulse control, Finance and Stochastics, 2 (1998), 85-114.
doi: 10.1007/s007800050034. |
[22] |
P. Li, M. Zhou and C. Yin,
Optimal reinsurance with both proportional and fixed costs, Statistics & Probability Letters, 106 (2015), 134-141.
doi: 10.1016/j.spl.2015.06.024. |
[23] |
J. Paulsen and H. Gjessing,
Ruin theory with stochastic return on investments, Advances in Applied Probability, 29 (1997), 965-985.
doi: 10.2307/1427849. |
[24] |
S. Peng, Backward stochastic differential equations-stochastic optimization theory and viscosity solutions of hjb equations, Topics on Stochastic Analysis, 85–138. Google Scholar |
[25] |
S. Richard,
Optimal impulse control of a diffusion process with both fixed and proportional costs of control, SIAM J. Control Optim., 15 (1977), 79-91.
doi: 10.1137/0315007. |
[26] |
M. Taksar,
Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.
doi: 10.1007/s001860050001. |
[27] |
Z. Wu and Z. Yu,
Dynamic programming principle for one kind of stochastic recursive optimal control problem and hamilton–jacobi–bellman equation, SIAM Journal on Control and Optimization, 47 (2008), 2616-2641.
doi: 10.1137/060671917. |
[28] |
Z. Wu and L. Zhang,
The corporate optimal portfolio and consumption choice problem in the real project with borrowing rate higher than deposit rate, Applied mathematics and computation, 175 (2006), 1596-1608.
doi: 10.1016/j.amc.2005.09.007. |
[29] |
J. Xiong, S. Zhang, H Zhao and X. Zeng,
Optimal proportional reinsurance and investment problem with jump-diffusion risk process under effect of inside information, Frontiers of Mathematics in China, 9 (2014), 965-982.
doi: 10.1007/s11464-014-0403-5. |
[30] |
H. Yang and L. Zhang,
Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.
doi: 10.1016/j.insmatheco.2005.06.009. |
[31] |
C. Yin and K. C. Yuen,
Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs, Journal of Industrial and Management Optimization, 11 (2015), 1247-1262.
doi: 10.3934/jimo.2015.11.1247. |
[32] |
S. Zhang,
Impulse stochastic control for the optimization of the dividend payments of the compound Poisson risk model perturbed by diffusion, Stochastic Analysis and Applications, 30 (2012), 642-661.
doi: 10.1080/07362994.2012.684324. |
[33] |
X. Zhang, M. Zhou and J. Y. Guo,
Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Applied Stochastic Models in Business and Industry, 23 (2007), 63-71.
doi: 10.1002/asmb.637. |


[1] |
Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021003 |
[2] |
Zhongbao Zhou, Yanfei Bai, Helu Xiao, Xu Chen. A non-zero-sum reinsurance-investment game with delay and asymmetric information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 909-936. doi: 10.3934/jimo.2020004 |
[3] |
Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020179 |
[4] |
Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020170 |
[5] |
Shan Liu, Hui Zhao, Ximin Rong. Time-consistent investment-reinsurance strategy with a defaultable security under ambiguous environment. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021015 |
[6] |
Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020178 |
[7] |
Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 |
[8] |
Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033 |
[9] |
José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271 |
[10] |
Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 |
[11] |
Haili Yuan, Yijun Hu. Optimal investment for an insurer under liquid reserves. Journal of Industrial & Management Optimization, 2021, 17 (1) : 339-355. doi: 10.3934/jimo.2019114 |
[12] |
Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036 |
[13] |
Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185 |
[14] |
Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020394 |
[15] |
Nan Zhang, Linyi Qian, Zhuo Jin, Wei Wang. Optimal stop-loss reinsurance with joint utility constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 841-868. doi: 10.3934/jimo.2020001 |
[16] |
Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020336 |
[17] |
Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127 |
[18] |
Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021023 |
[19] |
Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455 |
[20] |
Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360 |
2019 Impact Factor: 1.366
Tools
Article outline
Figures and Tables
[Back to Top]