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March  2011, 1(1): 21-40. doi: 10.3934/mcrf.2011.1.21

Numerical methods for dividend optimization using regime-switching jump-diffusion models

Zhuo Jin 1, , George Yin 1, and  Hailiang Yang 2,

1. 

Department of Mathematics, Wayne State University, Detroit, Michigan 48202, United States, United States
2. 

Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong

Received  August 2010 Revised  October 2010 Published  March 2011

This work develops numerical methods for finding optimal dividend policies to maximize the expected present value of dividend payout, where the surplus follows a regime-switching jump diffusion model and the switching is represented by a continuous-time Markov chain. To approximate the optimal dividend policies or optimal controls, we use Markov chain approximation techniques to construct a discrete-time controlled Markov chain with two components. Under simple conditions, we prove the convergence of the approximation sequence to the surplus process and the convergence of the approximation to the value function. Several examples are provided to demonstrate the performance of the algorithms.
Keywords: stochastic control., Jump diffusion, regime switching, dividend policy.
Mathematics Subject Classification: 93E20, 91B30, 91B70, 60H35, 65C05, 65C3.
Citation: Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control & Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21
References:
[1]

S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Math. and Economics, 20 (1997), 1-15. doi: 10.1016/S0167-6687(96)00017-0.  Google Scholar

[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, DFinance and Stochastics, 4 (2000), 299-324. doi: 10.1007/s007800050075.  Google Scholar

[3]

A. Cadenillas, T. Choulli, M. Taksar and L. Zhang, Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm, Mathematical Finance, 16 (2006), 181-202. doi: 10.1111/j.1467-9965.2006.00267.x.  Google Scholar

[4]

P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: a continuous-time model, Insurance: Math. Economics, 43 (2008), 456-465. doi: 10.1016/j.insmatheco.2008.09.001.  Google Scholar

[5]

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

[6]

R. J. Elliott and T. K. Siu, Robust optimal portfolio choice under Markovian regime-switching model, Methodology and Computing in Applied Probability, 11 (2009), 145-157. doi: 10.1007/s11009-008-9085-3.  Google Scholar

[7]

H. Gerber, Games of economic survival with discrete- and continuous-income processes, Operations Res., 20 (1972), 37-45. doi: 10.1287/opre.20.1.37.  Google Scholar

[8]

H. Gerber, "An Introduction to Mathematical Risk Theory," in "Huebner Foundation Monograph 8" Distributed by Richard D. Irwin, Homewood, IL, 1979.  Google Scholar

[9]

H. Gerber and E. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20.  Google Scholar

[10]

H. Gerber and E. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93.  Google Scholar

[11]

Z. Jiang and M. Pistorius, Optimal dividend distribution under Markov regime switching, Working paper, (2008). arXiv:0812.4978 Google Scholar

[12]

H. J. Kushner and P. Dupuis, "Numerical Methods for Stochastic Control Problems in Continuous Time," 2nd edition, Springer, New York, 2001.  Google Scholar

[13]

C. F. Lee and S. W. Forbes, Dividend policy, Equity value, and cost of capital estimates in the property and liability insurance industry, J. Risk Insurance, 47 (1980), 205-222. doi: 10.2307/252328.  Google Scholar

[14]

H. Schmidli, "Stochastic Control in Insurance," Springer Verlag, 2008.  Google Scholar

[15]

Q. S. Song, G. Yin and Z. Zhang, Numerical method for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42 (2006), 1147-1157. doi: 10.1016/j.automatica.2006.03.016.  Google Scholar

[16]

L. Sotomayor and A. Cadenillas, Classical, singular, and impulse stochastic control for the optimal dividend policy when there is regime switching, Preprint, (2008), Download at http://papers.ssrn.com/sol3/papers.cfm?abstract-id=1139444. Google Scholar

[17]

J. Wei, H. Yang and R. Wang, Optimal reinsurance and dividend strategies under the Markov-Modulated insurance risk model, Stochastic Analysis and Applications, 28 (2010), 1078-1105. doi: 10.1080/07362994.2010.515488.  Google Scholar

[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.  Google Scholar

[19]

G. Yin, Q. Zhang and G. Badowski, Discrete-time singularly perturbed Markov chains: Aggregation, occupation measures, and switching diffusion limit, Adv. Appl. Probab., 35 (2003), 449-476. doi: 10.1239/aap/1051201656.  Google Scholar

[20]

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

[21]

Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim., 40 (2001), 64-87. doi: 10.1137/S0363012999356325.  Google Scholar

[22]

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.  Google Scholar

show all references

References:
[1]

S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Math. and Economics, 20 (1997), 1-15. doi: 10.1016/S0167-6687(96)00017-0.  Google Scholar

[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, DFinance and Stochastics, 4 (2000), 299-324. doi: 10.1007/s007800050075.  Google Scholar

[3]

A. Cadenillas, T. Choulli, M. Taksar and L. Zhang, Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm, Mathematical Finance, 16 (2006), 181-202. doi: 10.1111/j.1467-9965.2006.00267.x.  Google Scholar

[4]

P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: a continuous-time model, Insurance: Math. Economics, 43 (2008), 456-465. doi: 10.1016/j.insmatheco.2008.09.001.  Google Scholar

[5]

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

[6]

R. J. Elliott and T. K. Siu, Robust optimal portfolio choice under Markovian regime-switching model, Methodology and Computing in Applied Probability, 11 (2009), 145-157. doi: 10.1007/s11009-008-9085-3.  Google Scholar

[7]

H. Gerber, Games of economic survival with discrete- and continuous-income processes, Operations Res., 20 (1972), 37-45. doi: 10.1287/opre.20.1.37.  Google Scholar

[8]

H. Gerber, "An Introduction to Mathematical Risk Theory," in "Huebner Foundation Monograph 8" Distributed by Richard D. Irwin, Homewood, IL, 1979.  Google Scholar

[9]

H. Gerber and E. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20.  Google Scholar

[10]

H. Gerber and E. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93.  Google Scholar

[11]

Z. Jiang and M. Pistorius, Optimal dividend distribution under Markov regime switching, Working paper, (2008). arXiv:0812.4978 Google Scholar

[12]

H. J. Kushner and P. Dupuis, "Numerical Methods for Stochastic Control Problems in Continuous Time," 2nd edition, Springer, New York, 2001.  Google Scholar

[13]

C. F. Lee and S. W. Forbes, Dividend policy, Equity value, and cost of capital estimates in the property and liability insurance industry, J. Risk Insurance, 47 (1980), 205-222. doi: 10.2307/252328.  Google Scholar

[14]

H. Schmidli, "Stochastic Control in Insurance," Springer Verlag, 2008.  Google Scholar

[15]

Q. S. Song, G. Yin and Z. Zhang, Numerical method for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42 (2006), 1147-1157. doi: 10.1016/j.automatica.2006.03.016.  Google Scholar

[16]

L. Sotomayor and A. Cadenillas, Classical, singular, and impulse stochastic control for the optimal dividend policy when there is regime switching, Preprint, (2008), Download at http://papers.ssrn.com/sol3/papers.cfm?abstract-id=1139444. Google Scholar

[17]

J. Wei, H. Yang and R. Wang, Optimal reinsurance and dividend strategies under the Markov-Modulated insurance risk model, Stochastic Analysis and Applications, 28 (2010), 1078-1105. doi: 10.1080/07362994.2010.515488.  Google Scholar

[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.  Google Scholar

[19]

G. Yin, Q. Zhang and G. Badowski, Discrete-time singularly perturbed Markov chains: Aggregation, occupation measures, and switching diffusion limit, Adv. Appl. Probab., 35 (2003), 449-476. doi: 10.1239/aap/1051201656.  Google Scholar

[20]

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

[21]

Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim., 40 (2001), 64-87. doi: 10.1137/S0363012999356325.  Google Scholar

[22]

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.  Google Scholar

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