# American Institute of Mathematical Sciences

September  2019, 24(9): 5083-5105. doi: 10.3934/dcdsb.2019044

## optimal investment and dividend policy in an insurance company: A varied bound for dividend rates

 School of Mathematical Sciences, Tongji University, Shanghai 200092, China

Received  July 2018 Revised  October 2018 Published  February 2019

In this paper we consider an optimal dividend problem for an insurance company whose surplus process evolves a classical ${\rm Cram\acute{e}r}$-Lundberg process. We impose a varied bound over the dividend rate to raise the dividend payment at a acceptable survival probability. Our objective is to find a strategy consisting of both investment and dividend payment which maximizes the cumulative expected discounted dividend payment until the ruin time. We show that the optimal value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a given boundary condition. We characterize the optimal value function as the smallest viscosity supersolution of the HJB equation. We introduce a method to construct the potential solution of our problem and give a verification theorem to check its optimality. Finally we show some numerical results.

Citation: Yiling Chen, Baojun Bian. optimal investment and dividend policy in an insurance company: A varied bound for dividend rates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5083-5105. doi: 10.3934/dcdsb.2019044
##### References:
 [1] H. Albrecher and S. Thonhauser, Optimality results for dividend problems in insurance, RACSAM-Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A, Matematicas, 103 (2009), 295-320. doi: 10.1007/BF03191909. Google Scholar [2] H. Albrecher and A. Cani, Risk theory with affine dividend payment strategies, Number Theory-Diophantine Problems, Uniform Distribution and Applications, Springer, Cham, (2017), 25–60. Google Scholar [3] B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251. doi: 10.1080/10920277.2009.10597549. Google Scholar [4] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative L$\rm \acute{e}$vy process, The Annals of Applied Probability, 17 (2007), 156-180. doi: 10.1214/105051606000000709. Google Scholar [5] P. Azcue and N. Muler, Stochastic Optimization in Insurance, Springer, New York, 2014. doi: 10.1007/978-1-4939-0995-7. Google Scholar [6] P. Azcue and N. Muler, Optimal Reinsurance and dividend distribution policy in the Cram$\rm \acute{e}$r-Lundberg model, Mathematical Finance, 15 (2005), 261-308. doi: 10.1111/j.0960-1627.2005.00220.x. Google Scholar [7] P. Azcue and N. Muler, Optimal investment policy and dividend payment strategy in an insurance company, The annals of applied probability, 20 (2010), 1253-1302. doi: 10.1214/09-AAP643. Google Scholar [8] P. Azcue and N. Muler, Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates, Insurance: Mathematics and Economics, 51 (2012), 26-42. doi: 10.1016/j.insmatheco.2012.02.011. Google Scholar [9] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8. Google Scholar [10] M. G. Crandall and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American mathematical society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [11] J. Eisenberg, Optimal dividends under a stochastic interest rate, Insurance: Mathematics and Economics, 65 (2015), 259-266. doi: 10.1016/j.insmatheco.2015.10.007. Google Scholar [12] B. De Finetti, Su una impostazione alternativ della teoria collettiva del risichio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. Google Scholar [13] P. A. Forsyth and G. Labahn, Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, Journal of Computational Finance, 11 (2007), 1-43. doi: 10.21314/JCF.2007.163. Google Scholar [14] H. U. Gerber and E. S. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93. doi: 10.1080/10920277.2006.10596249. Google Scholar [15] F. Hubalek and W. Schachermayer, Optimizing expected utility of dividend payments for a Brownian risk process and a peculiar nonlinear ODE, Insurance: Mathematics and Economics, 34 (2004), 193-225. doi: 10.1016/j.insmatheco.2003.12.001. Google Scholar [16] N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cram$\rm \acute{e}$r-Lundberg model with capital injections, Insurance: Mathematics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013. Google Scholar [17] J. Paulsen, Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs, Advances in Applied Probability, 39 (2007), 669-689. doi: 10.1017/S0001867800001993. Google Scholar [18] N. Scheer and H. Schmidli, Optimal dividend strategies in a Cram$\rm \acute{e}$r-Lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92. doi: 10.1007/s13385-011-0007-3. Google Scholar [19] H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008. Google Scholar [20] Q. Song and C. Zhu, On singular control problems with state constraints and regime-switching: A viscosity solution approach, Automatica, 70 (2016), 66-73. doi: 10.1016/j.automatica.2016.03.017. Google Scholar [21] M. I. 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

show all references

##### References:
 [1] H. Albrecher and S. Thonhauser, Optimality results for dividend problems in insurance, RACSAM-Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A, Matematicas, 103 (2009), 295-320. doi: 10.1007/BF03191909. Google Scholar [2] H. Albrecher and A. Cani, Risk theory with affine dividend payment strategies, Number Theory-Diophantine Problems, Uniform Distribution and Applications, Springer, Cham, (2017), 25–60. Google Scholar [3] B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251. doi: 10.1080/10920277.2009.10597549. Google Scholar [4] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative L$\rm \acute{e}$vy process, The Annals of Applied Probability, 17 (2007), 156-180. doi: 10.1214/105051606000000709. Google Scholar [5] P. Azcue and N. Muler, Stochastic Optimization in Insurance, Springer, New York, 2014. doi: 10.1007/978-1-4939-0995-7. Google Scholar [6] P. Azcue and N. Muler, Optimal Reinsurance and dividend distribution policy in the Cram$\rm \acute{e}$r-Lundberg model, Mathematical Finance, 15 (2005), 261-308. doi: 10.1111/j.0960-1627.2005.00220.x. Google Scholar [7] P. Azcue and N. Muler, Optimal investment policy and dividend payment strategy in an insurance company, The annals of applied probability, 20 (2010), 1253-1302. doi: 10.1214/09-AAP643. Google Scholar [8] P. Azcue and N. Muler, Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates, Insurance: Mathematics and Economics, 51 (2012), 26-42. doi: 10.1016/j.insmatheco.2012.02.011. Google Scholar [9] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8. Google Scholar [10] M. G. Crandall and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American mathematical society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [11] J. Eisenberg, Optimal dividends under a stochastic interest rate, Insurance: Mathematics and Economics, 65 (2015), 259-266. doi: 10.1016/j.insmatheco.2015.10.007. Google Scholar [12] B. De Finetti, Su una impostazione alternativ della teoria collettiva del risichio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. Google Scholar [13] P. A. Forsyth and G. Labahn, Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, Journal of Computational Finance, 11 (2007), 1-43. doi: 10.21314/JCF.2007.163. Google Scholar [14] H. U. Gerber and E. S. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93. doi: 10.1080/10920277.2006.10596249. Google Scholar [15] F. Hubalek and W. Schachermayer, Optimizing expected utility of dividend payments for a Brownian risk process and a peculiar nonlinear ODE, Insurance: Mathematics and Economics, 34 (2004), 193-225. doi: 10.1016/j.insmatheco.2003.12.001. Google Scholar [16] N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cram$\rm \acute{e}$r-Lundberg model with capital injections, Insurance: Mathematics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013. Google Scholar [17] J. Paulsen, Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs, Advances in Applied Probability, 39 (2007), 669-689. doi: 10.1017/S0001867800001993. Google Scholar [18] N. Scheer and H. Schmidli, Optimal dividend strategies in a Cram$\rm \acute{e}$r-Lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92. doi: 10.1007/s13385-011-0007-3. Google Scholar [19] H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008. Google Scholar [20] Q. Song and C. Zhu, On singular control problems with state constraints and regime-switching: A viscosity solution approach, Automatica, 70 (2016), 66-73. doi: 10.1016/j.automatica.2016.03.017. Google Scholar [21] M. I. 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
(a) The optimal value function. (b) The optimal dividend payment strategy. (c) The the optimal investment policy. (d) The first order derivative of $V(x)$
Optimal value function for $p = 4$ and different $g$ restrictions
Survival probability function under optimal strategy for $p = 2$ and different $g$ restrictions
 [1] Fengjun Wang, Qingling Zhang, Bin Li, Wanquan Liu. Optimal investment strategy on advertisement in duopoly. Journal of Industrial & Management Optimization, 2016, 12 (2) : 625-636. doi: 10.3934/jimo.2016.12.625 [2] 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 [3] 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 [4] 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 [5] Jiaqin Wei, Danping Li, Yan Zeng. Robust optimal consumption-investment strategy with non-exponential discounting. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-24. doi: 10.3934/jimo.2018147 [6] Dingjun Yao, Rongming Wang, Lin Xu. Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1235-1259. doi: 10.3934/jimo.2014.10.1235 [7] 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 [8] Ka Wo Lau, Yue Kuen Kwok. Optimal execution strategy of liquidation. Journal of Industrial & Management Optimization, 2006, 2 (2) : 135-144. doi: 10.3934/jimo.2006.2.135 [9] Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315 [10] Linyi Qian, Lyu Chen, Zhuo Jin, Rongming Wang. Optimal liability ratio and dividend payment strategies under catastrophic risk. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1443-1461. doi: 10.3934/jimo.2018015 [11] Guibin Lu, Qiying Hu, Youying Zhou, Wuyi Yue. Optimal execution strategy with an endogenously determined sales period. Journal of Industrial & Management Optimization, 2005, 1 (3) : 289-304. doi: 10.3934/jimo.2005.1.289 [12] 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 [13] Xuanhua Peng, Wen Su, Zhimin Zhang. On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-20. doi: 10.3934/jimo.2019038 [14] Yujing Wang, Changjun Yu, Kok Lay Teo. A new computational strategy for optimal control problem with a cost on changing control. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 339-364. doi: 10.3934/naco.2016016 [15] Jingang Zhai, Guangmao Jiang, Jianxiong Ye. Optimal dilution strategy for a microbial continuous culture based on the biological robustness. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 59-69. doi: 10.3934/naco.2015.5.59 [16] Yuxue Li, Maozhu Jin, Peiyu Ren, Zhixue Liao. Research on the optimal initial shunt strategy of Jiuzhaigou based on the optimization model. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1239-1249. doi: 10.3934/dcdss.2015.8.1239 [17] Heman Shakeri, Faryad Darabi Sahneh, Caterina Scoglio, Pietro Poggi-Corradini, Victor M. Preciado. Optimal information dissemination strategy to promote preventive behaviors in multilayer epidemic networks. Mathematical Biosciences & Engineering, 2015, 12 (3) : 609-623. doi: 10.3934/mbe.2015.12.609 [18] Siyu Liu, Xue Yang, Yingjie Bi, Yong Li. Dynamic behavior and optimal scheduling for mixed vaccination strategy with temporary immunity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1469-1483. doi: 10.3934/dcdsb.2018216 [19] Lin Xu, Dingjun Yao, Gongpin Cheng. Optimal investment and dividend for an insurer under a Markov regime switching market with high gain tax. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-32. doi: 10.3934/jimo.2018154 [20] Jingzhen Liu, Ka-Fai Cedric Yiu, Tak Kuen Siu, Wai-Ki Ching. Optimal insurance in a changing economy. Mathematical Control & Related Fields, 2014, 4 (2) : 187-202. doi: 10.3934/mcrf.2014.4.187

2018 Impact Factor: 1.008