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

[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

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

[1]

Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043

[2]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[3]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053

[4]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[5]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

[6]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[7]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021003

[8]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017

[9]

Yiling Chen, Baojun Bian. Optimal dividend policy in an insurance company with contagious arrivals of claims. Mathematical Control & Related Fields, 2021, 11 (1) : 1-22. doi: 10.3934/mcrf.2020024

[10]

Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283

[11]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[12]

Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021010

[13]

Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020179

[14]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

[15]

Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381

[16]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[17]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[18]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[19]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398

[20]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (52)
  • HTML views (40)
  • Cited by (0)

Other articles
by authors

[Back to Top]