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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

* Corresponding author: Shuaiqi Zhang

* Corresponding author: Xin Zhang

Received  October 2018 Revised  August 2019 Published  March 2021 Early access  December 2019

Fund Project: The first author is supported by the National Natural Science Foundation of China (grants 11771079, 11371020), the second author is supported by Southern University of Science and Technology Start up fund Y01286120 and National Natural Science Foundation of China (grants 61873325, 11831010), and the third author is supported by the National Natural Science Foundation of China (grant 11501129)

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.

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. AsmussenB. 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. CadenillasT. ChoulliM. 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. ChoulliM. 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. ChoulliM. 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. HarrisonT. 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. LiM. 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. XiongS. ZhangH 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. ZhangM. 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. AsmussenB. 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. CadenillasT. ChoulliM. 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. ChoulliM. 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. ChoulliM. 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. HarrisonT. 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. LiM. 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. XiongS. ZhangH 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. ZhangM. 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 1.  The figure of $ V'(x) $
Figure 2.  The figure of the value function $ V(x) $
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