# American Institute of Mathematical Sciences

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. Asmussen, B. 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. Avram, Z. 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. Bai, J. 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. 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. [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. 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. [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. 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. [18] H. Zhao, X. 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.

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##### References:
 [1] S. Asmussen, B. 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. Avram, Z. 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. Bai, J. 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. 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. [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. 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. [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. 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. [18] H. Zhao, X. 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.
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