# American Institute of Mathematical Sciences

October 2018, 14(4): 1323-1348. doi: 10.3934/jimo.2018009

## Optimal investment and dividend payment strategies with debt management and reinsurance

 1 School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai 201620, China 2 Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia 3 School of Statistics, Faculty of Economics and Management, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China

* Corresponding author

Received  August 2016 Revised  October 2017 Published  January 2018

Fund Project: This work was supported by Program of Shanghai Subject Chief Scientist (14XD1401600), the 111 Project (B14019), National Natural Science Foundation of China (11601157,11601320,11571113,11231005,11501211), Research Grants Council of the Hong Kong Special Administrative Region (project No. HKU 17330816) and Faculty Research Grant by The University of Melbourne

This paper derives the optimal debt ratio, investment and dividend payment strategies for an insurance company. The surplus process is jointly determined by the reinsurance strategies, debt levels, investment portfolios and unanticipated shocks. The objective is to maximize the total expected discounted utility of dividend payments in finite-time period subject to three control variables. The utility functions are chosen as the logarithmic and power utility functions. Using dynamic programming principle, the value function is the solution of a second-order nonlinear Hamilton-Jacobi-Bellman equation. The explicit solution of the value function is derived and the corresponding optimal debt ratio, investment and dividend payment strategies are obtained. In addition, the investment borrowing constraint, dividend payment constraint and impacts of reinsurance policies are considered and their impacts on the optimal strategies are analyzed. Further, to incorporating the interest rate risk, the problem is studied under a stochastic interest rate model.

Citation: 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
##### 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] S. Asmussen, B. Høgaard 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. [3] 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. [4] 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. [5] Y. C. Chi and H. Meng, Optimal reinsurance arrangements in the presence of two reinsurers, Scandinavian Actuarial Journal, 5 (2014), 424-438. [6] T. Choulli, M. Taksar and X. Y. Zhou, Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quant. Finance, 1 (2001), 573-596. doi: 10.1088/1469-7688/1/6/301. [7] B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [8] W. H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM Journal of Control and Optimization, 43 (2004), 502-531. doi: 10.1137/S0363012902419060. [9] H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20. doi: 10.1080/10920277.2004.10596125. [10] H. U. Gerber and E. S. W. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93. doi: 10.1080/10920277.2006.10596249. [11] Z. Jin, H. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica, 49 (2013), 2317-2329. doi: 10.1016/j.automatica.2013.04.043. [12] Z. Jin, H. Yang and G. Yin, Optimal debt ratio and dividend payment strategies with reinsurance, Insurance: Mathematics and Economics, 64 (2015), 351-363. doi: 10.1016/j.insmatheco.2015.07.005. [13] N. Kulenko and H. Schimidli, An optimal dividend strategy in a Craḿer Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013. [14] Z. F. Li, Y. Zeng and Y. Z. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203. doi: 10.1016/j.insmatheco.2011.09.002. [15] H. Meng and T. K. Siu, Optimal mixed impulse-equity insurance control problem with reinsurance, SIAM Journal on Control and Optimization, 49 (2011), 254-279. doi: 10.1137/090773167. [16] C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York, 1992. [17] Stein and L. Jerome, Stochastic Optimal Control and the U. S. Financial Debt Crisis Springer, New York, 2012. [18] J. Wei, H. Yang and R. Wang, Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching, Journal of Optimization Theory and Applications, 147 (2010), 358-377. doi: 10.1007/s10957-010-9726-x. [19] D. Yao, H. Yang and R. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European Journal of Operational Research, 211 (2011), 568-576. doi: 10.1016/j.ejor.2011.01.015. [20] G. Yin, H. Jin and Z. Jin, Numerical methods for portfolio selection with bounded constraints, J. Computational Appl. Math., 233 (2009), 564-581. doi: 10.1016/j.cam.2009.08.055. [21] X. Y. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583. [22] M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modeling, 29 (2012), 198-207. doi: 10.1016/j.econmod.2011.09.007. [23] J. Zhu, Dividend optimization for a regime-switching diffusion model with restricted dividend rates, ASTIN Bulletin, 44 (2014), 459-494. doi: 10.1017/asb.2014.2.

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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] S. Asmussen, B. Høgaard 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. [3] 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. [4] 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. [5] Y. C. Chi and H. Meng, Optimal reinsurance arrangements in the presence of two reinsurers, Scandinavian Actuarial Journal, 5 (2014), 424-438. [6] T. Choulli, M. Taksar and X. Y. Zhou, Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quant. Finance, 1 (2001), 573-596. doi: 10.1088/1469-7688/1/6/301. [7] B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [8] W. H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM Journal of Control and Optimization, 43 (2004), 502-531. doi: 10.1137/S0363012902419060. [9] H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20. doi: 10.1080/10920277.2004.10596125. [10] H. U. Gerber and E. S. W. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93. doi: 10.1080/10920277.2006.10596249. [11] Z. Jin, H. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica, 49 (2013), 2317-2329. doi: 10.1016/j.automatica.2013.04.043. [12] Z. Jin, H. Yang and G. Yin, Optimal debt ratio and dividend payment strategies with reinsurance, Insurance: Mathematics and Economics, 64 (2015), 351-363. doi: 10.1016/j.insmatheco.2015.07.005. [13] N. Kulenko and H. Schimidli, An optimal dividend strategy in a Craḿer Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013. [14] Z. F. Li, Y. Zeng and Y. Z. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203. doi: 10.1016/j.insmatheco.2011.09.002. [15] H. Meng and T. K. Siu, Optimal mixed impulse-equity insurance control problem with reinsurance, SIAM Journal on Control and Optimization, 49 (2011), 254-279. doi: 10.1137/090773167. [16] C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York, 1992. [17] Stein and L. Jerome, Stochastic Optimal Control and the U. S. Financial Debt Crisis Springer, New York, 2012. [18] J. Wei, H. Yang and R. Wang, Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching, Journal of Optimization Theory and Applications, 147 (2010), 358-377. doi: 10.1007/s10957-010-9726-x. [19] D. Yao, H. Yang and R. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European Journal of Operational Research, 211 (2011), 568-576. doi: 10.1016/j.ejor.2011.01.015. [20] G. Yin, H. Jin and Z. Jin, Numerical methods for portfolio selection with bounded constraints, J. Computational Appl. Math., 233 (2009), 564-581. doi: 10.1016/j.cam.2009.08.055. [21] X. Y. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583. [22] M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modeling, 29 (2012), 198-207. doi: 10.1016/j.econmod.2011.09.007. [23] J. Zhu, Dividend optimization for a regime-switching diffusion model with restricted dividend rates, ASTIN Bulletin, 44 (2014), 459-494. doi: 10.1017/asb.2014.2.
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