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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  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 & 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.  Google Scholar [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.  Google Scholar [3] S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, Singapore: World Scientific, 2010. doi: 10.1142/9789814282536.  Google Scholar [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.  Google Scholar [5] A. Bensoussan and J. Lions, Impulse Control and Quasivariational Inequalities, $\mu$, Gauthier-Villars, Montrouge, 1984, Translated from the French by J. M. Cole.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scandinavian Actuarial Journal, 1998 (1998), 166-180.   Google Scholar [19] M. Jeanblanc-Picque and A. Shiryaev, Optimization of the flow of dividends, Russian Mathematical Surveys, 50 (1995), 257-277.  doi: 10.1070/RM1995v050n02ABEH002054.  Google Scholar [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.  Google Scholar [21] R. Korn, Portfolio optimisation with strictly positive transaction costs and impulse control, Finance and Stochastics, 2 (1998), 85-114.  doi: 10.1007/s007800050034.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, Singapore: World Scientific, 2010. doi: 10.1142/9789814282536.  Google Scholar [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.  Google Scholar [5] A. Bensoussan and J. Lions, Impulse Control and Quasivariational Inequalities, $\mu$, Gauthier-Villars, Montrouge, 1984, Translated from the French by J. M. Cole.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scandinavian Actuarial Journal, 1998 (1998), 166-180.   Google Scholar [19] M. Jeanblanc-Picque and A. Shiryaev, Optimization of the flow of dividends, Russian Mathematical Surveys, 50 (1995), 257-277.  doi: 10.1070/RM1995v050n02ABEH002054.  Google Scholar [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.  Google Scholar [21] R. Korn, Portfolio optimisation with strictly positive transaction costs and impulse control, Finance and Stochastics, 2 (1998), 85-114.  doi: 10.1007/s007800050034.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
The figure of $V'(x)$
The figure of the value function $V(x)$
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