American Institute of Mathematical Sciences

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

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

Jiaqin Wei 1, , Zhuo Jin 2,, and  Hailiang Yang 3,

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  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.

Keywords: Stochastic control, dividend strategies, liability constraint, regime-switching.
Mathematics Subject Classification: Primary: 90B50, 93E20; Secondary: 91G80.
Citation: Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial & 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.  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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443.   Google Scholar

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R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer: Berlin, Germany, 1995.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. Liptser and A. Shiryaev, Nonlinear filtering of diffusion type Markov processes, Transactions of the Steklov Mathematics Institute, 104 (1968), 135-180.   Google Scholar

[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.  Google Scholar

[22]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.  Google Scholar

[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.   Google Scholar

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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443.   Google Scholar

[9]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer: Berlin, Germany, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. Liptser and A. Shiryaev, Nonlinear filtering of diffusion type Markov processes, Transactions of the Steklov Mathematics Institute, 104 (1968), 135-180.   Google Scholar

[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.  Google Scholar

[22]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.  Google Scholar

[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.   Google Scholar

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