American Institute of Mathematical Sciences

Advanced
  • 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

Dingjun Yao 1, , Rongming Wang 2, and  Lin Xu 3,

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.
Keywords: restricted dividend rate, dual jump-diffusion model, dividend payment, capital injection, Optimal strategy, fixed cost..
Mathematics Subject Classification: Primary: 93E20, 91G50; Secondary: 91G8.
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 & 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.  doi: 10.1007/BF03191909.  Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1214/105051606000000709.  Google Scholar

[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.  doi: 10.1080/03461230802591098.  Google Scholar

[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.  doi: 10.1111/j.1467-9965.2006.00267.x.  Google Scholar

[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.  doi: 10.1007/s00186-010-0312-7.  Google Scholar

[8]

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

[9]

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

[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.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[11]

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

[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.  doi: 10.1137/070691632.  Google Scholar

[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.  doi: 10.1016/j.insmatheco.2012.08.004.  Google Scholar

[14]

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

[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.  doi: 10.1007/s13385-011-0007-3.  Google Scholar

[16]

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

[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.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[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.  doi: 10.1016/j.ejor.2011.01.015.  Google Scholar

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.  doi: 10.1007/BF03191909.  Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1214/105051606000000709.  Google Scholar

[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.  doi: 10.1080/03461230802591098.  Google Scholar

[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.  doi: 10.1111/j.1467-9965.2006.00267.x.  Google Scholar

[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.  doi: 10.1007/s00186-010-0312-7.  Google Scholar

[8]

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

[9]

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

[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.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[11]

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

[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.  doi: 10.1137/070691632.  Google Scholar

[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.  doi: 10.1016/j.insmatheco.2012.08.004.  Google Scholar

[14]

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

[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.  doi: 10.1007/s13385-011-0007-3.  Google Scholar

[16]

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

[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.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[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.  doi: 10.1016/j.ejor.2011.01.015.  Google Scholar

[1]

Chuancun Yin, Kam Chuen Yuen. Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. Journal of Industrial & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019047
[9]

Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control & Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003
[10]

Xuanhua Peng, Wen Su, Zhimin Zhang. On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-20. doi: 10.3934/jimo.2019038
[11]

Yiling Chen, Baojun Bian. optimal investment and dividend policy in an insurance company: A varied bound for dividend rates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5083-5105. doi: 10.3934/dcdsb.2019044
[12]

Ishak Alia. A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion. Mathematical Control & Related Fields, 2019, 9 (3) : 541-570. doi: 10.3934/mcrf.2019025
[13]

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 & Management Optimization, 2019, 15 (1) : 293-318. doi: 10.3934/jimo.2018044
[14]

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 & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2019072
[15]

Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control & Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020
[16]

Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132
[17]

Zhimin Zhang. On a risk model with randomized dividend-decision times. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1041-1058. doi: 10.3934/jimo.2014.10.1041
[18]

Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100
[19]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[20]

Yiling Chen, Baojun Bian. Optimal dividend policy in an insurance company with contagious arrivals of claims. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020024

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences