American Institute of Mathematical Sciences

Advanced
October  2015, 11(4): 1247-1262. doi: 10.3934/jimo.2015.11.1247

Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs

Chuancun Yin 1, and  Kam Chuen Yuen 2,

1. 

School of Statistics, Qufu Normal University, Shandong 273165, China
2. 

Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China

Received  April 2014 Revised  February 2015 Published  March 2015

In this paper, we study the optimal control problem for a company whose surplus process evolves as an upward jump diffusion with random return on investment. Three types of practical optimization problems faced by a company that can control its liquid reserves by paying dividends and injecting capital. In the first problem, we consider the classical dividend problem without capital injections. The second problem aims at maximizing the expected discounted dividend payments minus the expected discounted costs of capital injections over strategies with positive surplus at all times. The third problem has the same objective as the second one, but without the constraints on capital injections. Under the assumption of proportional transaction costs, we identify the value function and the optimal strategies for any distribution of gains.
Keywords: dual model, stochastic control., optimal dividend strategy, HJB equation, jump-diffusion, Barrier strategy.
Mathematics Subject Classification: Primary: 93E20, 91G80; Secondary: 60J7.
Citation: 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
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1992.

[2]

S. Asmussen, F. Avram and M. R. Pistorius, Russian and American put options under exponential phase-type Lévy models, Stochastic Processes and their Applications, 109 (2004), 79-111. doi: 10.1016/j.spa.2003.07.005.

[3]

B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251. doi: 10.1080/10920277.2009.10597549.

[4]

B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance: Mathematics and Economics, 52 (2013), 98-113. doi: 10.1016/j.insmatheco.2012.10.008.

[5]

B. Avanzi and H. U. Gerber, Optimal dividends in the dual model with diffusion, ASTIN Bulletin, 38 (2008), 653-667. doi: 10.2143/AST.38.2.2033357.

[6]

B. Avanzi, H. U. Gerber and E. S. W. Shiu, Optimal dividends in the dual model, Insurance: Mathematics and Economics, 41 (2007), 111-123. doi: 10.1016/j.insmatheco.2006.10.002.

[7]

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.

[8]

B. Avanzi, V. Tu and B. Wong, On optimal periodic dividend strategies in the dual model with diffusion, Insurance: Mathematics and Economics, 55 (2014), 210-224. doi: 10.1016/j.insmatheco.2014.01.005.

[9]

P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Mathematical Finance, 15 (2005), 261-308. doi: 10.1111/j.0960-1627.2005.00220.x.

[10]

E. Bayraktar and M. Egami, Optimizing venture capital investments in a jump diffusion model, Mathematical Methods of Operations Research, 67 (2008), 21-42. doi: 10.1007/s00186-007-0181-x.

[11]

E. Bayraktar, A. E. Kyprianou and K. Yamazaki, On optimal dividends in the dual model, ASTIN Bulletin, 43 (2013), 359-372. doi: 10.1017/asb.2013.17.

[12]

E. Bayraktar, A. E. Kyprianou and K. Yamazaki, Optimal dividends in the dual model under transaction costs, Insurance: Mathematics and Economics, 54 (2014), 133-143. doi: 10.1016/j.insmatheco.2013.11.007.

[13]

E. C. K. Cheung and S. Drekic, Dividend moments in the dual model: Exact and approximate approaches, ASTIN Bulletin, 38 (2008), 399-422. doi: 10.2143/AST.38.2.2033347.

[14]

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.

[15]

H. Dai, Z. Liu and N. Luan, Optimal financing and dividend control in the dual model, Mathematical and Computer Modelling, 53 (2011), 1921-1928. doi: 10.1016/j.mcm.2011.01.019.

[16]

B. De Finetti, Su un'impostazion alternativa dell teoria collecttiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443.

[17]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics, Springer-Verlag, New York, 1993.

[18]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: Mathematics and Economics, 44 (2009), 88-94. doi: 10.1016/j.insmatheco.2008.10.001.

[19]

S. Jaschke, A note on the inhomogeneous linear stochastic differential equation, Insurance: Mathematics and Economics, 32 (2003), 461-464. doi: 10.1016/S0167-6687(03)00134-3.

[20]

N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cramér-Lundberg model with capital injections, Insurance: Mathematics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013.

[21]

K. Miyasawa, An economic survival game, Journal of the Operations Research Society of Japan, 4 (1962), 95-113.

[22]

H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008.

[23]

D. J. Yao, H. L. Yang and R. M. 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.

[24]

D. J. Yao, H. L. Yang and R. W. 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.

[25]

D. J. Yao, R. W. Wang and L. Xu, Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model, Journal of Industrial and Management Optimization, 10 (2014), 1235-1259. doi: 10.3934/jimo.2014.10.1235.

[26]

C. C. Yin and Y. Z. Wen, Optimal dividends problem with a terminal value for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773. doi: 10.1016/j.insmatheco.2013.09.019.

[27]

C. C. Yin and Y. Z. Wen, An extension of Paulsen-Gjessing's risk model with stochastic return on investments, Insurance: Mathematics and Economics, 52 (2013), 469-476. doi: 10.1016/j.insmatheco.2013.02.014.

[28]

C. C. Yin, Y. Z. Wen and Y. X. Zhao, On the optimal dividend problem for a spectrally positive Lévy process, ASTIN Bulletin, 44 (2014), 635-651. doi: 10.1017/asb.2014.12.

[29]

Z. M. Zhang, On a risk model with randomized dividend-decision times, Journal of Industrial and Management Optimization, 10 (2014), 1041-1058. doi: 10.3934/jimo.2014.10.1041.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1992.

[2]

S. Asmussen, F. Avram and M. R. Pistorius, Russian and American put options under exponential phase-type Lévy models, Stochastic Processes and their Applications, 109 (2004), 79-111. doi: 10.1016/j.spa.2003.07.005.

[3]

B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251. doi: 10.1080/10920277.2009.10597549.

[4]

B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance: Mathematics and Economics, 52 (2013), 98-113. doi: 10.1016/j.insmatheco.2012.10.008.

[5]

B. Avanzi and H. U. Gerber, Optimal dividends in the dual model with diffusion, ASTIN Bulletin, 38 (2008), 653-667. doi: 10.2143/AST.38.2.2033357.

[6]

B. Avanzi, H. U. Gerber and E. S. W. Shiu, Optimal dividends in the dual model, Insurance: Mathematics and Economics, 41 (2007), 111-123. doi: 10.1016/j.insmatheco.2006.10.002.

[7]

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.

[8]

B. Avanzi, V. Tu and B. Wong, On optimal periodic dividend strategies in the dual model with diffusion, Insurance: Mathematics and Economics, 55 (2014), 210-224. doi: 10.1016/j.insmatheco.2014.01.005.

[9]

P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Mathematical Finance, 15 (2005), 261-308. doi: 10.1111/j.0960-1627.2005.00220.x.

[10]

E. Bayraktar and M. Egami, Optimizing venture capital investments in a jump diffusion model, Mathematical Methods of Operations Research, 67 (2008), 21-42. doi: 10.1007/s00186-007-0181-x.

[11]

E. Bayraktar, A. E. Kyprianou and K. Yamazaki, On optimal dividends in the dual model, ASTIN Bulletin, 43 (2013), 359-372. doi: 10.1017/asb.2013.17.

[12]

E. Bayraktar, A. E. Kyprianou and K. Yamazaki, Optimal dividends in the dual model under transaction costs, Insurance: Mathematics and Economics, 54 (2014), 133-143. doi: 10.1016/j.insmatheco.2013.11.007.

[13]

E. C. K. Cheung and S. Drekic, Dividend moments in the dual model: Exact and approximate approaches, ASTIN Bulletin, 38 (2008), 399-422. doi: 10.2143/AST.38.2.2033347.

[14]

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.

[15]

H. Dai, Z. Liu and N. Luan, Optimal financing and dividend control in the dual model, Mathematical and Computer Modelling, 53 (2011), 1921-1928. doi: 10.1016/j.mcm.2011.01.019.

[16]

B. De Finetti, Su un'impostazion alternativa dell teoria collecttiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443.

[17]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics, Springer-Verlag, New York, 1993.

[18]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: Mathematics and Economics, 44 (2009), 88-94. doi: 10.1016/j.insmatheco.2008.10.001.

[19]

S. Jaschke, A note on the inhomogeneous linear stochastic differential equation, Insurance: Mathematics and Economics, 32 (2003), 461-464. doi: 10.1016/S0167-6687(03)00134-3.

[20]

N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cramér-Lundberg model with capital injections, Insurance: Mathematics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013.

[21]

K. Miyasawa, An economic survival game, Journal of the Operations Research Society of Japan, 4 (1962), 95-113.

[22]

H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008.

[23]

D. J. Yao, H. L. Yang and R. M. 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.

[24]

D. J. Yao, H. L. Yang and R. W. 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.

[25]

D. J. Yao, R. W. Wang and L. Xu, Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model, Journal of Industrial and Management Optimization, 10 (2014), 1235-1259. doi: 10.3934/jimo.2014.10.1235.

[26]

C. C. Yin and Y. Z. Wen, Optimal dividends problem with a terminal value for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773. doi: 10.1016/j.insmatheco.2013.09.019.

[27]

C. C. Yin and Y. Z. Wen, An extension of Paulsen-Gjessing's risk model with stochastic return on investments, Insurance: Mathematics and Economics, 52 (2013), 469-476. doi: 10.1016/j.insmatheco.2013.02.014.

[28]

C. C. Yin, Y. Z. Wen and Y. X. Zhao, On the optimal dividend problem for a spectrally positive Lévy process, ASTIN Bulletin, 44 (2014), 635-651. doi: 10.1017/asb.2014.12.

[29]

Z. M. Zhang, On a risk model with randomized dividend-decision times, Journal of Industrial and Management Optimization, 10 (2014), 1041-1058. doi: 10.3934/jimo.2014.10.1041.

[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]

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
[3]

Ishak Alia. A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion. Mathematical Control and Related Fields, 2019, 9 (3) : 541-570. doi: 10.3934/mcrf.2019025
[4]

Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022048
[5]

Hassan Tahir, Asaf Khan, Anwarud Din, Amir Khan, Gul Zaman. Optimal control strategy for an age-structured SIR endemic model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2535-2555. doi: 10.3934/dcdss.2021054
[6]

Sheng Li, Wei Yuan, Peimin Chen. Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022068
[7]

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 and Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072
[8]

Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057
[9]

Xinhong Zhang, Qing Yang. Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3155-3175. doi: 10.3934/dcdsb.2021177
[10]

Haiyang Wang, Zhen Wu. Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation. Mathematical Control and Related Fields, 2015, 5 (3) : 651-678. doi: 10.3934/mcrf.2015.5.651
[11]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[12]

Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control and Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21
[13]

Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control and Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003
[14]

Jiongmin Yong. Time-inconsistent optimal control problems and the equilibrium HJB equation. Mathematical Control and Related Fields, 2012, 2 (3) : 271-329. doi: 10.3934/mcrf.2012.2.271
[15]

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
[16]

Dingjun Yao, Hailiang Yang, Rongming Wang. Optimal financing and dividend strategies in a dual model with proportional costs. Journal of Industrial and Management Optimization, 2010, 6 (4) : 761-777. doi: 10.3934/jimo.2010.6.761
[17]

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
[18]

Xuanhua Peng, Wen Su, Zhimin Zhang. On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1967-1986. doi: 10.3934/jimo.2019038
[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]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (110)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy