# 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  September 2019 Early access  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 and 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. [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. [3] B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251.  doi: 10.1080/10920277.2009.10597549. [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. [5] P. Azcue and N. Muler, Stochastic Optimization in Insurance, Springer, New York, 2014. doi: 10.1007/978-1-4939-0995-7. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008. [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. [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.

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##### 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. [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. [3] B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251.  doi: 10.1080/10920277.2009.10597549. [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. [5] P. Azcue and N. Muler, Stochastic Optimization in Insurance, Springer, New York, 2014. doi: 10.1007/978-1-4939-0995-7. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008. [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. [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.
(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
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