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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

 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.

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##### 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.
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