January  2018, 14(1): 371-395. doi: 10.3934/jimo.2017051

Optimal dividend and capital injection strategy with excess-of-loss reinsurance and transaction costs

1. 

School of Economics, Nanjing University of Finance and Economics, Nanjing 210023, China

2. 

School of Statistics, East China Normal University, Shanghai 200062, China

3. 

School of Finance, Nanjing University of Finance and Economics, Nanjing 210023, China

* Corresponding author

Received  July 2015 Revised  December 2016 Published  January 2018 Early access  June 2017

Fund Project: The authors are grateful to the two anonymous referees for their valuable suggestions. This work was supported by the National Natural Science Foundation of China (11571113,11231005,71471081,71671082), the Humanities and Social Sciences Project of the Ministry Education of China (15YJC910008), the Program of Shanghai Subject Chief Scientist (14XD1401600), the 111 Project (B14019), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (15KJB110009).

This article deals with an optimal dividend, reinsurance and capital injection control problem in the diffusion risk model. Under the objective of maximizing the insurance company's value, we aim at finding the joint optimal control strategy. We assume that there exist both the fixed and proportional costs in control processes and the excess-of-loss reinsurance is "expensive". We derive the closed-form solutions of the value function and optimal strategy by using stochastic control methods. Some economic interpretations of the obtained results are also given.

Citation: 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
References:
[1]

S. AsmussenB. H$\phi$gaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of-loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.  doi: 10.1007/s007800050075.

[2]

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

[3]

L. BaiJ. Guo and H. Zhang, Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes, Quantitative Finance, 10 (2010), 1163-1172.  doi: 10.1080/14697680902968005.

[4]

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.

[5]

B. De Finetti, Su un'impostzione alternativa della teoria collettiva del rischio, In: Transactions of the XVth International Congress of Actuaries, New York: Congrès International d'Actuaires, 2 (1957), 433-443.

[6] J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991.  doi: 10.1007/978-1-4613-9058-9.
[7]

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.

[8]

C. Hipp and M. Taksar, Optimal non-proportional reinsurance control, Insurance: Mathematics and Economics, 47 (2010), 246-254.  doi: 10.1016/j.insmatheco.2010.04.001.

[9]

N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cram$\acute{e}$r-Lundberg model with capital injections, Insurance: Mathematics and Economics, 43 (2008), 270-278.  doi: 10.1016/j.insmatheco.2008.05.013.

[10]

W. Liu and Y. Hu, Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy, Statistics Probability Letters, 84 (2014), 121-130.  doi: 10.1016/j.spl.2013.09.034.

[11]

A. L$\phi$kka 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.

[12]

H. Meng and T. Siu, On optimal reinsurance, dividend and reinvestment strategies, Economic Modelling, 28 (2011), 211-218. 

[13]

H. Meng and X. Zhang, Optimal risk control for the excess of loss reinsurance policies, ASTIN Bulletin, 40 (2010), 179-197.  doi: 10.2143/AST.40.1.2049224.

[14]

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.

[15]

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.

[16]

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.

[17]

D. YaoH. 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.

[18]

H. ZhaoX. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance: Mathematics and Economics, 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004.

[19]

M. Zhou and K. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207. 

show all references

References:
[1]

S. AsmussenB. H$\phi$gaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of-loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.  doi: 10.1007/s007800050075.

[2]

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

[3]

L. BaiJ. Guo and H. Zhang, Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes, Quantitative Finance, 10 (2010), 1163-1172.  doi: 10.1080/14697680902968005.

[4]

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.

[5]

B. De Finetti, Su un'impostzione alternativa della teoria collettiva del rischio, In: Transactions of the XVth International Congress of Actuaries, New York: Congrès International d'Actuaires, 2 (1957), 433-443.

[6] J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991.  doi: 10.1007/978-1-4613-9058-9.
[7]

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.

[8]

C. Hipp and M. Taksar, Optimal non-proportional reinsurance control, Insurance: Mathematics and Economics, 47 (2010), 246-254.  doi: 10.1016/j.insmatheco.2010.04.001.

[9]

N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cram$\acute{e}$r-Lundberg model with capital injections, Insurance: Mathematics and Economics, 43 (2008), 270-278.  doi: 10.1016/j.insmatheco.2008.05.013.

[10]

W. Liu and Y. Hu, Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy, Statistics Probability Letters, 84 (2014), 121-130.  doi: 10.1016/j.spl.2013.09.034.

[11]

A. L$\phi$kka 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.

[12]

H. Meng and T. Siu, On optimal reinsurance, dividend and reinvestment strategies, Economic Modelling, 28 (2011), 211-218. 

[13]

H. Meng and X. Zhang, Optimal risk control for the excess of loss reinsurance policies, ASTIN Bulletin, 40 (2010), 179-197.  doi: 10.2143/AST.40.1.2049224.

[14]

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.

[15]

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.

[16]

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.

[17]

D. YaoH. 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.

[18]

H. ZhaoX. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance: Mathematics and Economics, 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004.

[19]

M. Zhou and K. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207. 

[1]

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

[2]

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

[3]

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

[4]

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

[5]

Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial and Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008

[6]

Nan Zhang, Linyi Qian, Zhuo Jin, Wei Wang. Optimal stop-loss reinsurance with joint utility constraints. Journal of Industrial and Management Optimization, 2021, 17 (2) : 841-868. doi: 10.3934/jimo.2020001

[7]

Xia Han, Zhibin Liang, Yu Yuan, Caibin Zhang. Optimal per-loss reinsurance and investment to minimize the probability of drawdown. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021145

[8]

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

[9]

Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial and Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044

[10]

Xiaoyu Xing, Caixia Geng. Optimal investment-reinsurance strategy in the correlated insurance and financial markets. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021120

[11]

Qianru li, Weida chen, Yongming zhang. Optimal production and emission reduction policies for a remanufacturing firm considering deferred payment strategy. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2475-2503. doi: 10.3934/jimo.2020078

[12]

Yujing Wang, Changjun Yu, Kok Lay Teo. A new computational strategy for optimal control problem with a cost on changing control. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 339-364. doi: 10.3934/naco.2016016

[13]

Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial and Management Optimization, 2022, 18 (1) : 75-93. doi: 10.3934/jimo.2020143

[14]

Hiroshi Konno, Tomokazu Hatagi. Index-plus-alpha tracking under concave transaction cost. Journal of Industrial and Management Optimization, 2005, 1 (1) : 87-98. doi: 10.3934/jimo.2005.1.87

[15]

Wenyan Zhuo, Honglin Yang, Leopoldo Eduardo Cárdenas-Barrón, Hong Wan. Loss-averse supply chain decisions with a capital constrained retailer. Journal of Industrial and Management Optimization, 2021, 17 (2) : 711-732. doi: 10.3934/jimo.2019131

[16]

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

[17]

Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080

[18]

Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial and Management Optimization, 2022, 18 (2) : 795-823. doi: 10.3934/jimo.2020179

[19]

Ka Wo Lau, Yue Kuen Kwok. Optimal execution strategy of liquidation. Journal of Industrial and Management Optimization, 2006, 2 (2) : 135-144. doi: 10.3934/jimo.2006.2.135

[20]

Bing-Bing Cao, Zai-Jing Gong, Tian-Hui You. Stackelberg pricing policy in dyadic capital-constrained supply chain considering bank's deposit and loan based on delay payment scheme. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2855-2887. doi: 10.3934/jimo.2020098

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (250)
  • HTML views (595)
  • Cited by (3)

Other articles
by authors

[Back to Top]