March  2017, 7(1): 1-19. doi: 10.3934/mcrf.2017001

Optimal size of business and dividend strategy in a nonlinear model with refinancing and liquidation value

1. 

School of Statistics, Faculty of Economics and Management, East China Normal University, Shanghai 200241, China

2. 

School of Mathematics and Computer Sciences, Anhui Normal University, Wuhu 241000, China

* Corresponding author

Received  July 2015 Revised  February 2016 Published  December 2016

Fund Project: This work was supported by National Natural Science Foundation of China (11571113,11231005,11201006), Program of Shanghai Subject Chief Scientist(14XD1401600), the 111 Project (B14019), Education of Humanities and Social Science Fund Project (12YJC910012)

This paper investigates the optimal control problem with a nonlinear capital process attributed to internal competition factors. Suppose that the company can control its capital process by paying dividend, refinancing and changing the size of business. The transaction costs generated by the control processes as well as the liquidation value at ruin are considered. We aim at seeking the optimal control strategies for maximizing the company's value. The results show that the company should expand the business scale when the current capital increases. The refinancing may only happen at the moments when, and only when, the capital is null. The dividends should be paid out according to barrier strategy if the dividend rate is unconstrained or threshold strategy if the dividend rate is bounded, respectively.

Citation: Gongpin Cheng, Lin Xu. Optimal size of business and dividend strategy in a nonlinear model with refinancing and liquidation value. Mathematical Control & Related Fields, 2017, 7 (1) : 1-19. doi: 10.3934/mcrf.2017001
References:
[1]

A. CadenillasT. ChoulliM. 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

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M. ChenX. Peng and J. Guo, Optimal dividend problem with a nonlinear regular-singular stochastic control, Insurance: Mathematics and Economics, 52 (2013), 448-456.  doi: 10.1016/j.insmatheco.2013.02.010.  Google Scholar

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T. ChoulliM. Taksar and X. Zhou, A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979.  doi: 10.1137/S0363012900382667.  Google Scholar

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W. Fleming and H. Soner, Controlled Markov Process and Viscosity Solutions, London, 1993, Springer-Verlag.  Google Scholar

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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.  Google Scholar

[6]

H. Guan and Z. Liang, Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs, Insurance: Mathematics and Economics, 54 (2014), 109-122.  doi: 10.1016/j.insmatheco.2013.11.003.  Google Scholar

[7]

X. Guo, Some risk management problems for firms with internal competition and debt, Journal of Applied Probability, 39 (2002), 55-69.  doi: 10.1017/S0021900200021501.  Google Scholar

[8]

X. GuoJ. Liu and X. Zhou, A constrained non-linear regular-singular stochastic control problem, with applications, Stochastic Processes and their Applications, 109 (2004), 167-187.  doi: 10.1016/j.spa.2003.09.008.  Google Scholar

[9]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: Mathematics and Economics, 44 (2009), 88-94.  doi: 10.1016/j.insmatheco.2008.10.001.  Google Scholar

[10]

B. Hϕgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066.  Google Scholar

[11]

B. Hϕgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 9 (2004), 153-182.   Google Scholar

[12]

Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.  doi: 10.1016/j.insmatheco.2012.02.005.  Google Scholar

[13]

P. Lions and A. Sznitman, Stochastic differential equations with reflecting boundary conditions, Communications on Pure and Applied Mathematics, 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408.  Google Scholar

[14]

H. MengT. Siu and H. Yang, Optimal dividends with debts and nonlinear insurance risk processes, Insurance: Mathematics and Economics, 53 (2013), 110-121.  doi: 10.1016/j.insmatheco.2013.04.008.  Google Scholar

[15]

J. Paulsen, Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs, SIAM Journal on Control and Optimization, 47 (2008), 2201-2226.  doi: 10.1137/070691632.  Google Scholar

[16]

X. PengM. Chen and J. Guo, Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.  doi: 10.1016/j.insmatheco.2012.08.004.  Google Scholar

[17]

S. Sethi and M. Taksar, Optimal financing of a corporation subject to random returns, Mathematical Finance, 12 (2002), 155-172.  doi: 10.1111/1467-9965.t01-2-02002.  Google Scholar

[18]

M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.  doi: 10.1007/s001860050001.  Google Scholar

[19]

M. Taksar, Dependence of the optimal risk control decisions on the terminal value for a financial corporation, Annals of Operations Research, 98 (2000), 89-99.  doi: 10.1023/A:1019239920624.  Google Scholar

[20]

M. Taksar and X. Zhou, Optimal risk and dividend control for a company with a debt liability, Insurance: Mathematics and Economics, 22 (1998), 105-122.  doi: 10.1016/S0167-6687(98)00012-2.  Google Scholar

[21]

J. Xu and M. Zhou, Optimal risk control and dividend distribution policies for a diffusion model with terminal value, Mathematical and Computer Modelling, 56 (2012), 180-190.  doi: 10.1016/j.mcm.2011.12.041.  Google Scholar

[22]

D. YaoH. Yang and R. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.  doi: 10.1016/j.econmod.2013.10.026.  Google Scholar

[23]

Y. ZhaoR. Wang and D. Yao, Optimal dividend and equity issuance in the perturbed dual model under a penalty for ruin, Communications in Statistics -Theory and Methods, 45 (2016), 365-384.  doi: 10.1080/03610926.2013.810269.  Google Scholar

[24]

M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.  doi: 10.1016/j.econmod.2011.09.007.  Google Scholar

show all references

References:
[1]

A. CadenillasT. ChoulliM. 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

[2]

M. ChenX. Peng and J. Guo, Optimal dividend problem with a nonlinear regular-singular stochastic control, Insurance: Mathematics and Economics, 52 (2013), 448-456.  doi: 10.1016/j.insmatheco.2013.02.010.  Google Scholar

[3]

T. ChoulliM. Taksar and X. Zhou, A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979.  doi: 10.1137/S0363012900382667.  Google Scholar

[4]

W. Fleming and H. Soner, Controlled Markov Process and Viscosity Solutions, London, 1993, Springer-Verlag.  Google Scholar

[5]

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.  Google Scholar

[6]

H. Guan and Z. Liang, Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs, Insurance: Mathematics and Economics, 54 (2014), 109-122.  doi: 10.1016/j.insmatheco.2013.11.003.  Google Scholar

[7]

X. Guo, Some risk management problems for firms with internal competition and debt, Journal of Applied Probability, 39 (2002), 55-69.  doi: 10.1017/S0021900200021501.  Google Scholar

[8]

X. GuoJ. Liu and X. Zhou, A constrained non-linear regular-singular stochastic control problem, with applications, Stochastic Processes and their Applications, 109 (2004), 167-187.  doi: 10.1016/j.spa.2003.09.008.  Google Scholar

[9]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: Mathematics and Economics, 44 (2009), 88-94.  doi: 10.1016/j.insmatheco.2008.10.001.  Google Scholar

[10]

B. Hϕgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066.  Google Scholar

[11]

B. Hϕgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 9 (2004), 153-182.   Google Scholar

[12]

Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.  doi: 10.1016/j.insmatheco.2012.02.005.  Google Scholar

[13]

P. Lions and A. Sznitman, Stochastic differential equations with reflecting boundary conditions, Communications on Pure and Applied Mathematics, 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408.  Google Scholar

[14]

H. MengT. Siu and H. Yang, Optimal dividends with debts and nonlinear insurance risk processes, Insurance: Mathematics and Economics, 53 (2013), 110-121.  doi: 10.1016/j.insmatheco.2013.04.008.  Google Scholar

[15]

J. Paulsen, Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs, SIAM Journal on Control and Optimization, 47 (2008), 2201-2226.  doi: 10.1137/070691632.  Google Scholar

[16]

X. PengM. Chen and J. Guo, Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.  doi: 10.1016/j.insmatheco.2012.08.004.  Google Scholar

[17]

S. Sethi and M. Taksar, Optimal financing of a corporation subject to random returns, Mathematical Finance, 12 (2002), 155-172.  doi: 10.1111/1467-9965.t01-2-02002.  Google Scholar

[18]

M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.  doi: 10.1007/s001860050001.  Google Scholar

[19]

M. Taksar, Dependence of the optimal risk control decisions on the terminal value for a financial corporation, Annals of Operations Research, 98 (2000), 89-99.  doi: 10.1023/A:1019239920624.  Google Scholar

[20]

M. Taksar and X. Zhou, Optimal risk and dividend control for a company with a debt liability, Insurance: Mathematics and Economics, 22 (1998), 105-122.  doi: 10.1016/S0167-6687(98)00012-2.  Google Scholar

[21]

J. Xu and M. Zhou, Optimal risk control and dividend distribution policies for a diffusion model with terminal value, Mathematical and Computer Modelling, 56 (2012), 180-190.  doi: 10.1016/j.mcm.2011.12.041.  Google Scholar

[22]

D. YaoH. Yang and R. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.  doi: 10.1016/j.econmod.2013.10.026.  Google Scholar

[23]

Y. ZhaoR. Wang and D. Yao, Optimal dividend and equity issuance in the perturbed dual model under a penalty for ruin, Communications in Statistics -Theory and Methods, 45 (2016), 365-384.  doi: 10.1080/03610926.2013.810269.  Google Scholar

[24]

M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.  doi: 10.1016/j.econmod.2011.09.007.  Google Scholar

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