American Institute of Mathematical Sciences

Advanced
October  2010, 6(4): 761-777. doi: 10.3934/jimo.2010.6.761

Optimal financing and dividend strategies in a dual model with proportional costs

Dingjun Yao 1, , Hailiang Yang 2, and  Rongming Wang 3,

1. 

School of Finance, Nanjing University of Finance and Economics, Nanjing, 210046, China
2. 

Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong
3. 

School of Finance and Statistics, East China Normal University, Shanghai, 200241

Received  August 2009 Revised  April 2010 Published  September 2010

We consider the optimal control problem with dividend payments and issuance of equity in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. Assuming proportional transaction costs, we aim at finding optimal strategy which maximizes the expected present value of the dividends payout minus the discounted costs of issuing new equity before bankruptcy. By adopting some of the techniques and methodologies in L$\phi$kka and Zervos (2008), we construct two categories of suboptimal models, one is the ordinary dual model without issuance of equity, the other one assumes that, by issuing new equity, the company never goes bankrupt. We identify the value functions and the optimal strategies corresponding to the suboptimal models in two different cases. For exponentially distributed jump sizes, closed-form solutions are obtained.
Keywords: Hamilton-Jacobi-Bellman equation., The dual risk model, dividend payment, optimal strategy, proportional transaction costs, equity issuance.
Mathematics Subject Classification: Primary: 93E20, 49L20 Secondary: 91G8.
Citation: Dingjun Yao, Hailiang Yang, Rongming Wang. Optimal financing and dividend strategies in a dual model with proportional costs. Journal of Industrial & Management Optimization, 2010, 6 (4) : 761-777. doi: 10.3934/jimo.2010.6.761
References:
[1]

S. Asmussen, B. Høgaard 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.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

B. De Finetti, Su un'impostazione alternativa dell teoria colletiva del rischio, Transactions of the XV International Congress of Actuaries, 2 (1957), 433-443.  Google Scholar

[5]

Y. H. Dong and G. J. Wang, Ruin probability for renewal risk model with negative risk sums, Journal of Industrial and Management Optimization, 2 (2006), 229-236.  Google Scholar

[6]

W. H. Flemming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Springer-Verlag, NewYork, 1993.  Google Scholar

[7]

H. U. Gerber, Games of economic survival with discrete- and continuous-income processes, Operations Research, 20 (1972), 37-45. doi: 10.1287/opre.20.1.37.  Google Scholar

[8]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20.  Google Scholar

[9]

J. Grandell, "Aspects of Risk Theory," New York, Springer-Verlag, 1991.  Google Scholar

[10]

L. He and Z. X. Liang, Optimal financing and dividend control of the insurance company with proportional reinsurance strategy, Insurance: Mathematics and Economics, 42 (2008), 976-983. doi: 10.1016/j.insmatheco.2007.11.003.  Google Scholar

[11]

B. Høgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution strategy, Quantitative Finance, 4 (2004), 315-327. doi: 10.1088/1469-7688/4/3/007.  Google Scholar

[12]

M. Jeanblanc and A. N. Shiryaev, Optimization of the flow of dividends, Russian Mathematical Surveys, 50 (1995), 257-277. doi: 10.1070/RM1995v050n02ABEH002054.  Google Scholar

[13]

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

[14]

G. Lu, Q. Hu, Y. Zhou and W. Yue, Optimal execution strategy with an endogenously determined sales period, Journal of Industrial and Management Optimization, 1 (2005), 280-304.  Google Scholar

[15]

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

[16]

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

[17]

H. L. Seal, "Stochastic Theory of a Risk Business," Wiley, New York, 1969.  Google Scholar

[18]

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

[19]

L. Xu, R. M. Wang and D. J. Yao, On maximizing the expected terminal utility by investment and reinsurance, Journal of Industrial and Management Optimization, 4 (2008), 801-815.  Google Scholar

[20]

K. F. C. Yiu, S. Y. Wang and K. L. Mak, Optimal portfolio under a value-at-risk constraint with applications to inventory control in supply chains, Journal of Industrial and Management Optimization, 4 (2008), 81-94. doi: 10.3934/jimo.2009.5.81.  Google Scholar

[21]

J. X. Zhu and H. L. Yang, Ruin probabilities of a dual Markov-modulated risk model, Communications in Statistics-Theory and Methods, 37 (2008), 3298-3307. doi: 10.1080/03610920802117080.  Google Scholar

show all references

References:
[1]

S. Asmussen, B. Høgaard 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.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

B. De Finetti, Su un'impostazione alternativa dell teoria colletiva del rischio, Transactions of the XV International Congress of Actuaries, 2 (1957), 433-443.  Google Scholar

[5]

Y. H. Dong and G. J. Wang, Ruin probability for renewal risk model with negative risk sums, Journal of Industrial and Management Optimization, 2 (2006), 229-236.  Google Scholar

[6]

W. H. Flemming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Springer-Verlag, NewYork, 1993.  Google Scholar

[7]

H. U. Gerber, Games of economic survival with discrete- and continuous-income processes, Operations Research, 20 (1972), 37-45. doi: 10.1287/opre.20.1.37.  Google Scholar

[8]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20.  Google Scholar

[9]

J. Grandell, "Aspects of Risk Theory," New York, Springer-Verlag, 1991.  Google Scholar

[10]

L. He and Z. X. Liang, Optimal financing and dividend control of the insurance company with proportional reinsurance strategy, Insurance: Mathematics and Economics, 42 (2008), 976-983. doi: 10.1016/j.insmatheco.2007.11.003.  Google Scholar

[11]

B. Høgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution strategy, Quantitative Finance, 4 (2004), 315-327. doi: 10.1088/1469-7688/4/3/007.  Google Scholar

[12]

M. Jeanblanc and A. N. Shiryaev, Optimization of the flow of dividends, Russian Mathematical Surveys, 50 (1995), 257-277. doi: 10.1070/RM1995v050n02ABEH002054.  Google Scholar

[13]

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

[14]

G. Lu, Q. Hu, Y. Zhou and W. Yue, Optimal execution strategy with an endogenously determined sales period, Journal of Industrial and Management Optimization, 1 (2005), 280-304.  Google Scholar

[15]

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

[16]

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

[17]

H. L. Seal, "Stochastic Theory of a Risk Business," Wiley, New York, 1969.  Google Scholar

[18]

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

[19]

L. Xu, R. M. Wang and D. J. Yao, On maximizing the expected terminal utility by investment and reinsurance, Journal of Industrial and Management Optimization, 4 (2008), 801-815.  Google Scholar

[20]

K. F. C. Yiu, S. Y. Wang and K. L. Mak, Optimal portfolio under a value-at-risk constraint with applications to inventory control in supply chains, Journal of Industrial and Management Optimization, 4 (2008), 81-94. doi: 10.3934/jimo.2009.5.81.  Google Scholar

[21]

J. X. Zhu and H. L. Yang, Ruin probabilities of a dual Markov-modulated risk model, Communications in Statistics-Theory and Methods, 37 (2008), 3298-3307. doi: 10.1080/03610920802117080.  Google Scholar

[1]

Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369
[2]

Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3767-3793. doi: 10.3934/cpaa.2021130
[3]

Chuancun Yin, Kam Chuen Yuen. Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1247-1262. doi: 10.3934/jimo.2015.11.1247
[4]

Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161
[5]

Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, 2021, 29 (5) : 3429-3447. doi: 10.3934/era.2021046
[6]

Xuhui Wang, Lei Hu. A new method to solve the Hamilton-Jacobi-Bellman equation for a stochastic portfolio optimization model with boundary memory. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021137
[7]

Daniele Castorina, Annalisa Cesaroni, Luca Rossi. On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1251-1263. doi: 10.3934/cpaa.2016.15.1251
[8]

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
[9]

Gongpin Cheng, Rongming Wang, Dingjun Yao. Optimal dividend and capital injection strategy with excess-of-loss reinsurance and transaction costs. Journal of Industrial & Management Optimization, 2018, 14 (1) : 371-395. doi: 10.3934/jimo.2017051
[10]

Linyi Qian, Lyu Chen, Zhuo Jin, Rongming Wang. Optimal liability ratio and dividend payment strategies under catastrophic risk. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1443-1461. doi: 10.3934/jimo.2018015
[11]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933
[12]

Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020179
[13]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223
[14]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
[15]

Gongpin Cheng, Lin Xu. Optimal size of business and dividend strategy in a nonlinear model with refinancing and liquidation value. Mathematical Control & Related Fields, 2017, 7 (1) : 1-19. doi: 10.3934/mcrf.2017001
[16]

Xuanhua Peng, Wen Su, Zhimin Zhang. On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1967-1986. doi: 10.3934/jimo.2019038
[17]

Alexander Melnikov, Hongxi Wan. CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 343-368. doi: 10.3934/puqr.2021017
[18]

Federica Masiero. Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 223-263. doi: 10.3934/dcds.2012.32.223
[19]

Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009
[20]

Zhen Wang, Sanyang Liu. Multi-period mean-variance portfolio selection with fixed and proportional transaction costs. Journal of Industrial & Management Optimization, 2013, 9 (3) : 643-657. doi: 10.3934/jimo.2013.9.643

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy