# American Institute of Mathematical Sciences

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

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

 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.
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. doi: 10.1016/S0167-6687(96)00017-0. [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. doi: 10.1007/s007800050075. [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. doi: 10.1111/j.1467-9965.2006.00267.x. [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. doi: 10.1016/j.insmatheco.2008.09.001. [5] B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio,, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433. [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. doi: 10.1007/s11009-008-9085-3. [7] H. Gerber, Games of economic survival with discrete- and continuous-income processes,, Operations Res., 20 (1972), 37. doi: 10.1287/opre.20.1.37. [8] H. Gerber, "An Introduction to Mathematical Risk Theory,", in, (1979). [9] H. Gerber and E. Shiu, Optimal dividends: Analysis with Brownian motion,, North American Actuarial Journal, 8 (2004), 1. [10] H. Gerber and E. Shiu, On optimal dividend strategies in the compound Poisson model,, North American Actuarial Journal, 10 (2006), 76. [11] Z. Jiang and M. Pistorius, Optimal dividend distribution under Markov regime switching,, Working paper, (2008). [12] H. J. Kushner and P. Dupuis, "Numerical Methods for Stochastic Control Problems in Continuous Time,", 2$^{nd}$ edition, (2001). [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. doi: 10.2307/252328. [14] H. Schmidli, "Stochastic Control in Insurance,", Springer Verlag, (2008). [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. doi: 10.1016/j.automatica.2006.03.016. [16] L. Sotomayor and A. Cadenillas, Classical, singular, and impulse stochastic control for the optimal dividend policy when there is regime switching,, Preprint, (2008). [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. doi: 10.1080/07362994.2010.515488. [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. doi: 10.1007/s10957-010-9726-x. [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. doi: 10.1239/aap/1051201656. [20] G. Yin and C. Zhu, "Hybrid Switching Diffusions: Properties and Applications,", Springer, (2010). doi: 10.1007/978-1-4419-1105-6. [21] Q. Zhang, Stock trading: An optimal selling rule,, SIAM J. Control Optim., 40 (2001), 64. doi: 10.1137/S0363012999356325. [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. doi: 10.1137/S0363012902405583.

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##### References:
 [1] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out,, Insurance: Math. and Economics, 20 (1997), 1. doi: 10.1016/S0167-6687(96)00017-0. [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. doi: 10.1007/s007800050075. [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. doi: 10.1111/j.1467-9965.2006.00267.x. [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. doi: 10.1016/j.insmatheco.2008.09.001. [5] B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio,, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433. [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. doi: 10.1007/s11009-008-9085-3. [7] H. Gerber, Games of economic survival with discrete- and continuous-income processes,, Operations Res., 20 (1972), 37. doi: 10.1287/opre.20.1.37. [8] H. Gerber, "An Introduction to Mathematical Risk Theory,", in, (1979). [9] H. Gerber and E. Shiu, Optimal dividends: Analysis with Brownian motion,, North American Actuarial Journal, 8 (2004), 1. [10] H. Gerber and E. Shiu, On optimal dividend strategies in the compound Poisson model,, North American Actuarial Journal, 10 (2006), 76. [11] Z. Jiang and M. Pistorius, Optimal dividend distribution under Markov regime switching,, Working paper, (2008). [12] H. J. Kushner and P. Dupuis, "Numerical Methods for Stochastic Control Problems in Continuous Time,", 2$^{nd}$ edition, (2001). [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. doi: 10.2307/252328. [14] H. Schmidli, "Stochastic Control in Insurance,", Springer Verlag, (2008). [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. doi: 10.1016/j.automatica.2006.03.016. [16] L. Sotomayor and A. Cadenillas, Classical, singular, and impulse stochastic control for the optimal dividend policy when there is regime switching,, Preprint, (2008). [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. doi: 10.1080/07362994.2010.515488. [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. doi: 10.1007/s10957-010-9726-x. [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. doi: 10.1239/aap/1051201656. [20] G. Yin and C. Zhu, "Hybrid Switching Diffusions: Properties and Applications,", Springer, (2010). doi: 10.1007/978-1-4419-1105-6. [21] Q. Zhang, Stock trading: An optimal selling rule,, SIAM J. Control Optim., 40 (2001), 64. doi: 10.1137/S0363012999356325. [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. doi: 10.1137/S0363012902405583.
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