# American Institute of Mathematical Sciences

October  2019, 15(4): 1965-1993. doi: 10.3934/jimo.2018132

## Optimal dividend policy with liability constraint under a hidden Markov regime-switching model

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

* Corresponding author: zjin@unimelb.edu.au

Received  March 2018 Revised  April 2018 Published  October 2019 Early access  August 2018

Fund Project: The first author is supported by National Natural Science Foundation of China under Grant (Nos. 11771466, 11601157, 11231005, 11571113), Program of Shanghai Subject Chief Scientist (14XD1401600), the 111 Project (B14019). The second and third author were supported in part by Research Grants Council of the Hong Kong Special Administrative Region (project No. HKU 17330816).

This paper deals with the optimal liability and dividend strategies for an insurance company in Markov regime-switching models. The objective is to maximize the total expected discounted utility of dividend payment in the infinite time horizon in the logarithm and power utility cases, respectively. The switching process, which is interpreted by a hidden Markov chain, is not completely observable. By using the technique of the Wonham filter, the partially observed system is converted to a completely observed one and the necessary information is recovered. The upper-lower solution method is used to show the existence of classical solution of the associated second-order nonlinear Hamilton-Jacobi-Bellman equation in the two-regime case. The explicit solution of the value function is derived and the corresponding optimal dividend policies and liability ratios are obtained. In the multi-regime case, a general setting of the Wonham filter is presented, and the value function is proved to be a viscosity solution of the associated system of Hamilton-Jacobi-Bellman equations.

Citation: Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132
##### References:
 [1] 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. [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] N. Baeuerle and U. Rieder, Portfolio optimization with jumps and unobservable intensity process, Mathematical Finance, 17 (2007), 205-224.  doi: 10.1111/j.1467-9965.2006.00300.x. [4] L. H. Bai, J. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance: Mathematics and Economics, 53 (2008), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008. [5] 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. [6] M. G. Crandell and P. Lions, Viscosity solution of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8. [7] M. G. Crandall, H. Ishii and P. Lions, User's guide to viscosity solutions of second order partial differential equations, American Mathematical Society. Bulletin. New Series, 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [8] B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [9] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer: Berlin, Germany, 1995. [10] R. J. Elliott and T. K. Siu, Option pricing and filtering with hidden Markov-modulated pure-jump processes, Applied Mathematical Finance, 20 (2013), 1-25.  doi: 10.1080/1350486X.2012.655929. [11] 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. [12] W. H. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, volume 25 of Stochastic Modelling and Applied Probability. Springer-Verlag, New York, NY, second edition, 2006. [13] H. U. Gerber, Games of economic survival with discrete and continuous income processes, Operations Research, 20 (1972), 37-45.  doi: 10.1287/opre.20.1.37. [14] 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. [15] H. U. Gerber and E. S. W. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 84. doi: 10.1080/10920277.2006.10597409. [16] U. Haussmann and J. Sass, Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain, Finance and Stochastics, 8 (2004), 553-577.  doi: 10.1007/s00780-004-0132-9. [17] 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. [18] R. Korn, T. K. Siu and A. Zhang, Asset allocation for a DC pension fund under regime switching environment, European Actuarial Journal, 1 (2011), 361-377.  doi: 10.1007/s13385-011-0021-5. [19] N. Kulenko and H. Schimidli, An optimal dividend strategy in a Craer Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278.  doi: 10.1016/j.insmatheco.2008.05.013. [20] R. Liptser and A. Shiryaev, Nonlinear filtering of diffusion type Markov processes, Transactions of the Steklov Mathematics Institute, 104 (1968), 135-180. [21] 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. [22] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [23] R. Rishel and K. Helmes, A variational inequality sufficient condition for optimal stopping with application to an optimal stock selling problem, SIAM Journal on Control and Optimization, 45 (2006), 580-598.  doi: 10.1137/050622699. [24] T. K. Siu, A BSDE approach to risk-based asset allocation of pension funds with regime switching, Annals of Operations Research, 201 (2012), 449-473.  doi: 10.1007/s10479-012-1211-5. [25] T. K. Siu, A stochastic flows approach for asset allocation with hidden economic environment, International Journal of Stochastic Analysis, 2015 (2015), Article ID 462524, 11 pages. doi: 10.1155/2015/462524. [26] Q. Song, R. Stockbridge and C. Zhu, On optimal harvesting problems in random environments, SIAM J. Control Optim., 49 (2011), 859-889.  doi: 10.1137/100797333. [27] K. Tran and G. Yin, Stochastic competitive Lotka-Volterra ecosystems under partial observation: Feedback controls for permanence and extinction, Journal of the Franklin Institute, 351 (2014), 4039-4064.  doi: 10.1016/j.jfranklin.2014.04.015. [28] J. Wei, R. Wang and H. Yang, Optimal surrender strategies for equity- indexed annuity investors with partial information, Statistics and Probability Letters, 82 (2012), 1251-1258.  doi: 10.1016/j.spl.2012.03.021. [29] W. Wonham, Some applications of stochastic differential equations to optimal non-linear filtering, Journal of the Society for Industrial and Applied Mathematics Series A Control, 2 (1965), 347-369. [30] Z. Yang, G. Yin and Q. Zhang, Mean-variance type controls involving a hidden Markov chain: Models and numerical approximation, IMA Journal of Mathematical Control and Information, 32 (2015), 867-888.  doi: 10.1093/imamci/dnu027. [31] H. Yang and G. Yin, Ruin probability for a model under Markovian switching regime, In T. L. Lai, H. Yang, and S. P. Yung, editors, Probability, Finance and Insurance, pages 206-217. World Scientific, River Edge, NJ, 2004. [32] 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. [33] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [34] L. Yu, Q. Zhang and G. Yin, Asset allocation for regime-switching market models under partial observation, Dynamic Systems and Applications, 23 (2014), 39-61.

show all references

##### References:
 [1] 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. [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] N. Baeuerle and U. Rieder, Portfolio optimization with jumps and unobservable intensity process, Mathematical Finance, 17 (2007), 205-224.  doi: 10.1111/j.1467-9965.2006.00300.x. [4] L. H. Bai, J. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance: Mathematics and Economics, 53 (2008), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008. [5] 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. [6] M. G. Crandell and P. Lions, Viscosity solution of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8. [7] M. G. Crandall, H. Ishii and P. Lions, User's guide to viscosity solutions of second order partial differential equations, American Mathematical Society. Bulletin. New Series, 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [8] B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [9] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer: Berlin, Germany, 1995. [10] R. J. Elliott and T. K. Siu, Option pricing and filtering with hidden Markov-modulated pure-jump processes, Applied Mathematical Finance, 20 (2013), 1-25.  doi: 10.1080/1350486X.2012.655929. [11] 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. [12] W. H. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, volume 25 of Stochastic Modelling and Applied Probability. Springer-Verlag, New York, NY, second edition, 2006. [13] H. U. Gerber, Games of economic survival with discrete and continuous income processes, Operations Research, 20 (1972), 37-45.  doi: 10.1287/opre.20.1.37. [14] 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. [15] H. U. Gerber and E. S. W. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 84. doi: 10.1080/10920277.2006.10597409. [16] U. Haussmann and J. Sass, Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain, Finance and Stochastics, 8 (2004), 553-577.  doi: 10.1007/s00780-004-0132-9. [17] 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. [18] R. Korn, T. K. Siu and A. Zhang, Asset allocation for a DC pension fund under regime switching environment, European Actuarial Journal, 1 (2011), 361-377.  doi: 10.1007/s13385-011-0021-5. [19] N. Kulenko and H. Schimidli, An optimal dividend strategy in a Craer Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278.  doi: 10.1016/j.insmatheco.2008.05.013. [20] R. Liptser and A. Shiryaev, Nonlinear filtering of diffusion type Markov processes, Transactions of the Steklov Mathematics Institute, 104 (1968), 135-180. [21] 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. [22] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [23] R. Rishel and K. Helmes, A variational inequality sufficient condition for optimal stopping with application to an optimal stock selling problem, SIAM Journal on Control and Optimization, 45 (2006), 580-598.  doi: 10.1137/050622699. [24] T. K. Siu, A BSDE approach to risk-based asset allocation of pension funds with regime switching, Annals of Operations Research, 201 (2012), 449-473.  doi: 10.1007/s10479-012-1211-5. [25] T. K. Siu, A stochastic flows approach for asset allocation with hidden economic environment, International Journal of Stochastic Analysis, 2015 (2015), Article ID 462524, 11 pages. doi: 10.1155/2015/462524. [26] Q. Song, R. Stockbridge and C. Zhu, On optimal harvesting problems in random environments, SIAM J. Control Optim., 49 (2011), 859-889.  doi: 10.1137/100797333. [27] K. Tran and G. Yin, Stochastic competitive Lotka-Volterra ecosystems under partial observation: Feedback controls for permanence and extinction, Journal of the Franklin Institute, 351 (2014), 4039-4064.  doi: 10.1016/j.jfranklin.2014.04.015. [28] J. Wei, R. Wang and H. Yang, Optimal surrender strategies for equity- indexed annuity investors with partial information, Statistics and Probability Letters, 82 (2012), 1251-1258.  doi: 10.1016/j.spl.2012.03.021. [29] W. Wonham, Some applications of stochastic differential equations to optimal non-linear filtering, Journal of the Society for Industrial and Applied Mathematics Series A Control, 2 (1965), 347-369. [30] Z. Yang, G. Yin and Q. Zhang, Mean-variance type controls involving a hidden Markov chain: Models and numerical approximation, IMA Journal of Mathematical Control and Information, 32 (2015), 867-888.  doi: 10.1093/imamci/dnu027. [31] H. Yang and G. Yin, Ruin probability for a model under Markovian switching regime, In T. L. Lai, H. Yang, and S. P. Yung, editors, Probability, Finance and Insurance, pages 206-217. World Scientific, River Edge, NJ, 2004. [32] 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. [33] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [34] L. Yu, Q. Zhang and G. Yin, Asset allocation for regime-switching market models under partial observation, Dynamic Systems and Applications, 23 (2014), 39-61.
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