Figure 1.
The figure of $ V'(x) $
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March
2021, 17(2): 981-999.
doi: 10.3934/jimo.2020008
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.
Keywords:
Mixed classical-impulse control,
Reinsurance-investment and Dividends,
Quasi-variational inequalities,
Transaction costs.
Mathematics Subject Classification:
Primary: 91B30; Secondary: 93E20.
Citation:
Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial and Management Optimization,
2021, 17
(2)
: 981-999.
doi: 10.3934/jimo.2020008
![]()
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. |
| [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. |
| [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. |
Figure 2.
The figure of the value function $ V(x) $
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