• Previous Article
    Optimal pricing policy for deteriorating items with preservation technology investment
  • JIMO Home
  • This Issue
  • Next Article
    Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem
October  2014, 10(4): 1235-1259. doi: 10.3934/jimo.2014.10.1235

Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model

1. 

School of Finance, The Center of Cooperative Innovation for Modern Service Industry, Nanjing University of Finance and Economics, Nanjing 210023, China

2. 

School of Finance and Statistics, Research Center of International Finance and Risk Management, East China Normal University, Shanghai 200241, China

3. 

School of Mathematics and Computer Sciences, Anhui Normal University, Wuhu, Anhui, 241003

Received  April 2013 Revised  October 2013 Published  February 2014

In the framework of dual risk model, Yao et al. [18](Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs. European Journal of Operational Research, 211, 568-576) show how to determine optimal dividend and capital injection strategy when the dividend rate is unrestricted and the bankruptcy is forbidden. In this paper, we further include constrain on dividend rate and allow for bankruptcy when it is in deficit. We seek the optimal strategy for maximizing the expected discounted dividends minus the discounted capital injections before bankruptcy. Explicit solutions for strategy and value function are obtained when income jumps follow a hyper-exponential distribution, the corresponding limit results are presented, some known results are extended.
Citation: Dingjun Yao, Rongming Wang, Lin Xu. Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1235-1259. doi: 10.3934/jimo.2014.10.1235
References:
[1]

H. Albrecher and S. Thonhauser, Optimality results for dividend problems in insurance, RACSAM Rev. R. Acad. Cien. Serie A. Mat., 103 (2009), 295-320. doi: 10.1007/BF03191909.

[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.

[3]

B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bulletin, 41 (2011), 611-644. doi: 10.2139/ssrn.1709174.

[4]

F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative Lévy process, The Annals of Applied Probability, 17 (2007), 156-180. doi: 10.1214/105051606000000709.

[5]

L. Bai and J. Guo, Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes, Scandinavian Actuarial Journal, 2010 (2010), 36-55. doi: 10.1080/03461230802591098.

[6]

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.

[7]

H. Dai, Z. Liu, and N. Luan, Optimal dividend strategies in a dual model with capital injections, Mathematical Methods of Operations Research, 72 (2010), 129-143. doi: 10.1007/s00186-010-0312-7.

[8]

A. Feldmann and W. Whitt, Fitting mixtures of exponentials to long-tail distributions to analyze network performance models, Performance Evaluation, 31 (1998), 245-279. doi: 10.1016/S0166-5316(97)00003-5.

[9]

N. Kulenko and H. Schimidli, Optimal dividend strategy in a Cramér-Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013.

[10]

A. Løkka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961. doi: 10.1016/j.insmatheco.2007.10.013.

[11]

A. C. Y. Ng, On a dual model with a dividend threshold, Insurance: Mathematics and Economics, 44 (2009), 315-324. doi: 10.1016/j.insmatheco.2008.11.011.

[12]

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.

[13]

X. Peng, M. 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.

[14]

S. P. 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.

[15]

N. Scheer and H. Schmidli, Optimal dividend strategies in a cramér-lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92. doi: 10.1007/s13385-011-0007-3.

[16]

S. Thonhauser and H. Albrecher, Dividend maximization under consideration of the time value of ruin, Insurance: Mathematics and Economics, 44 (2007), 163-184. doi: 10.1016/j.insmatheco.2006.10.013.

[17]

D. Yao, H. Yang and R. Wang, Optimal financing and dividend strategies in a dual model with proportional costs, Journal of Industrial and Management Optimization, 6 (2010), 761-777. doi: 10.3934/jimo.2010.6.761.

[18]

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.

show all references

References:
[1]

H. Albrecher and S. Thonhauser, Optimality results for dividend problems in insurance, RACSAM Rev. R. Acad. Cien. Serie A. Mat., 103 (2009), 295-320. doi: 10.1007/BF03191909.

[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.

[3]

B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bulletin, 41 (2011), 611-644. doi: 10.2139/ssrn.1709174.

[4]

F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative Lévy process, The Annals of Applied Probability, 17 (2007), 156-180. doi: 10.1214/105051606000000709.

[5]

L. Bai and J. Guo, Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes, Scandinavian Actuarial Journal, 2010 (2010), 36-55. doi: 10.1080/03461230802591098.

[6]

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.

[7]

H. Dai, Z. Liu, and N. Luan, Optimal dividend strategies in a dual model with capital injections, Mathematical Methods of Operations Research, 72 (2010), 129-143. doi: 10.1007/s00186-010-0312-7.

[8]

A. Feldmann and W. Whitt, Fitting mixtures of exponentials to long-tail distributions to analyze network performance models, Performance Evaluation, 31 (1998), 245-279. doi: 10.1016/S0166-5316(97)00003-5.

[9]

N. Kulenko and H. Schimidli, Optimal dividend strategy in a Cramér-Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013.

[10]

A. Løkka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961. doi: 10.1016/j.insmatheco.2007.10.013.

[11]

A. C. Y. Ng, On a dual model with a dividend threshold, Insurance: Mathematics and Economics, 44 (2009), 315-324. doi: 10.1016/j.insmatheco.2008.11.011.

[12]

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.

[13]

X. Peng, M. 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.

[14]

S. P. 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.

[15]

N. Scheer and H. Schmidli, Optimal dividend strategies in a cramér-lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92. doi: 10.1007/s13385-011-0007-3.

[16]

S. Thonhauser and H. Albrecher, Dividend maximization under consideration of the time value of ruin, Insurance: Mathematics and Economics, 44 (2007), 163-184. doi: 10.1016/j.insmatheco.2006.10.013.

[17]

D. Yao, H. Yang and R. Wang, Optimal financing and dividend strategies in a dual model with proportional costs, Journal of Industrial and Management Optimization, 6 (2010), 761-777. doi: 10.3934/jimo.2010.6.761.

[18]

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.

[1]

Chuancun Yin, Kam Chuen Yuen. Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1247-1262. doi: 10.3934/jimo.2015.11.1247

[2]

Gongpin Cheng, Rongming Wang, Dingjun Yao. Optimal dividend and capital injection strategy with excess-of-loss reinsurance and transaction costs. Journal of Industrial and Management Optimization, 2018, 14 (1) : 371-395. doi: 10.3934/jimo.2017051

[3]

Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control and Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21

[4]

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

[5]

Dingjun Yao, Hailiang Yang, Rongming Wang. Optimal financing and dividend strategies in a dual model with proportional costs. Journal of Industrial and Management Optimization, 2010, 6 (4) : 761-777. doi: 10.3934/jimo.2010.6.761

[6]

Linyi Qian, Lyu Chen, Zhuo Jin, Rongming Wang. Optimal liability ratio and dividend payment strategies under catastrophic risk. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1443-1461. doi: 10.3934/jimo.2018015

[7]

Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009

[8]

Linlin Tian, Xiaoyi Zhang, Yizhou Bai. Optimal dividend of compound poisson process under a stochastic interest rate. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2141-2157. doi: 10.3934/jimo.2019047

[9]

Yiling Chen, Baojun Bian. optimal investment and dividend policy in an insurance company: A varied bound for dividend rates. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5083-5105. doi: 10.3934/dcdsb.2019044

[10]

Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control and Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003

[11]

Xuanhua Peng, Wen Su, Zhimin Zhang. On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1967-1986. doi: 10.3934/jimo.2019038

[12]

Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022048

[13]

Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control and Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020

[14]

Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132

[15]

Zhimin Zhang. On a risk model with randomized dividend-decision times. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1041-1058. doi: 10.3934/jimo.2014.10.1041

[16]

Ishak Alia. A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion. Mathematical Control and Related Fields, 2019, 9 (3) : 541-570. doi: 10.3934/mcrf.2019025

[17]

Sheng Li, Wei Yuan, Peimin Chen. Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022068

[18]

Qing-Qing Yang, Wai-Ki Ching, Wanhua He, Tak-Kuen Siu. Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales. Journal of Industrial and Management Optimization, 2019, 15 (1) : 293-318. doi: 10.3934/jimo.2018044

[19]

Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072

[20]

Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (93)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]