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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), 295-320. 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), 1-15. 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), 611-644. 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), 156-180. 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), 36-55. 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), 181-202. 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), 129-143. 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), 245-279. 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), 270-278. 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), 954-961. 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), 315-324. 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), 2201-2226. 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), 576-585. 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), 155-172. 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), 57-92. 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), 163-184. 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), 761-777. 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), 568-576. 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), 295-320. 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), 1-15. 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), 611-644. 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), 156-180. 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), 36-55. 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), 181-202. 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), 129-143. 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), 245-279. 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), 270-278. 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), 954-961. 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), 315-324. 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), 2201-2226. 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), 576-585. 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), 155-172. 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), 57-92. 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), 163-184. 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), 761-777. 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), 568-576. doi: 10.1016/j.ejor.2011.01.015.  Google Scholar

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