American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The EPQ model with deteriorating items under two levels of trade credit in a supply chain system
  • JIMO Home
  • This Issue
  • Next Article
    Service investment and consumer returns policy in a vendor-managed inventory supply chain
April  2015, 11(2): 461-478. doi: 10.3934/jimo.2015.11.461

Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission

Dingjun Yao 1, , Rongming Wang 2, and  Lin Xu 3,

1. 

School of Finance, The Center of Cooperative Innovation for Modern Service Industry, Nanjing University of Finance and Economics, Nanjing 210023
2. 

School of Finance and Statistics, Research Center of International Finance and Risk Management, East China Normal University, Shanghai 200241
3. 

School of Mathematics and Computer Science, Anhui Normal University, Wuhu, Anhui, 241003

Received  August 2012 Revised  April 2014 Published  September 2014

We assume that the asset value process of some company is directly related to its stock price dynamics, which can be modeled by geometric Brownian motion. The company can control its asset by paying dividends and injecting capitals, of course both procedures imply proportional and fixed costs for the company. To maximize the expected present value of the dividend payments minus the capital injections until the time of bankruptcy, which is defined as the first time when the asset value falls below the regulation requirement $m $, we seek to find the joint optimal dividend payment and capital injection strategy. By solving the Quasi-variational inequalities, the optimal control problem is addressed, which depends on the parameters of the model and the costs. The sensitivities of transaction costs (such as tax, consulting fees) to the optimal strategy, the expected growth rate and volatility of the firm asset value are also examined, some interesting economic insights are included.
Keywords: Dividend payment, geometric Brownian motion, optimal strategy., transaction costs, capital injection.
Mathematics Subject Classification: Primary: 93E20; Secondary: 62P0.
Citation: Dingjun Yao, Rongming Wang, Lin Xu. Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission. Journal of Industrial & Management Optimization, 2015, 11 (2) : 461-478. doi: 10.3934/jimo.2015.11.461
References:
[1]

B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion,, ASTIN Bulletin, 41 (2011), 611.  doi: 10.2139/ssrn.1709174.  Google Scholar

[2]

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.  doi: 10.1214/105051606000000709.  Google Scholar

[3]

A. Cadenillas and F. Zapatero, Optimal central bank intervention in the foreign exchange market,, Journal of Economic Theory, 87 (1999), 218.  doi: 10.1006/jeth.1999.2523.  Google Scholar

[4]

J. Cai, H. U. Gerber and H. Yang, Optimal dividends in an Ornstein-Uhlenbeck type model with credit and debit interest,, North American Actuarial Journal, 10 (2006), 94.  doi: 10.1080/10920277.2006.10596250.  Google Scholar

[5]

H. U. Gerber and E. S. W. Shiu, Geometric Brownian motion models for assets and liabilities from pension funding to optimal dividends,, North American Actuarial Journal, 7 (2003), 37.  doi: 10.1080/10920277.2003.10596099.  Google Scholar

[6]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion,, North American Actuarial Journal, 8 (2004), 1.  doi: 10.1080/10920277.2004.10596125.  Google Scholar

[7]

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

[8]

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

[9]

K. S. Leung, Y. K. Kwok and S. Y. Leung, Finite-time dividend-ruin models,, Insurance: Mathematics and Economics, 42 (2008), 154.  doi: 10.1016/j.insmatheco.2007.01.014.  Google Scholar

[10]

P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions,, Communications on Pure and Applied Mathematics, 37 (1984), 511.  doi: 10.1002/cpa.3160370408.  Google Scholar

[11]

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.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[12]

H. Meng and T. Siu, On optimal reinsurance, dividend and reinvestment strategies,, Economic Modelling, 28 (2011), 211.  doi: 10.1016/j.econmod.2010.09.009.  Google Scholar

[13]

M. Ohnishi and M. Tsujimura, An impulse control of a geometric Brownian motion with quadratic costs,, European Journal of Operational Research, 168 (2006), 311.  doi: 10.1016/j.ejor.2004.07.006.  Google Scholar

[14]

J. Paulsen, Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs,, Advances in Applied Probability, 39 (2007), 669.  doi: 10.1239/aap/1189518633.  Google Scholar

[15]

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.  doi: 10.1137/070691632.  Google Scholar

[16]

S. P. Sethi and M. Taksar, Optimal financing of a corporation subject to random retures,, Mathematical Finance, 12 (2002), 155.  doi: 10.1111/1467-9965.t01-2-02002.  Google Scholar

[17]

S. E. Shreve, J. P. Lehoczky and D. P. Gaver, Optimal consumption for general diffusions with absorbing and reflecting barriers,, SIAM Journal on Control and Optimization, 22 (1984), 55.  doi: 10.1137/0322005.  Google Scholar

[18]

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.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[19]

D. Yao, H. Yang and R. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle,, Economic Modelling, 37 (2014), 53.  doi: 10.1016/j.econmod.2013.10.026.  Google Scholar

show all references

References:
[1]

B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion,, ASTIN Bulletin, 41 (2011), 611.  doi: 10.2139/ssrn.1709174.  Google Scholar

[2]

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.  doi: 10.1214/105051606000000709.  Google Scholar

[3]

A. Cadenillas and F. Zapatero, Optimal central bank intervention in the foreign exchange market,, Journal of Economic Theory, 87 (1999), 218.  doi: 10.1006/jeth.1999.2523.  Google Scholar

[4]

J. Cai, H. U. Gerber and H. Yang, Optimal dividends in an Ornstein-Uhlenbeck type model with credit and debit interest,, North American Actuarial Journal, 10 (2006), 94.  doi: 10.1080/10920277.2006.10596250.  Google Scholar

[5]

H. U. Gerber and E. S. W. Shiu, Geometric Brownian motion models for assets and liabilities from pension funding to optimal dividends,, North American Actuarial Journal, 7 (2003), 37.  doi: 10.1080/10920277.2003.10596099.  Google Scholar

[6]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion,, North American Actuarial Journal, 8 (2004), 1.  doi: 10.1080/10920277.2004.10596125.  Google Scholar

[7]

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

[8]

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

[9]

K. S. Leung, Y. K. Kwok and S. Y. Leung, Finite-time dividend-ruin models,, Insurance: Mathematics and Economics, 42 (2008), 154.  doi: 10.1016/j.insmatheco.2007.01.014.  Google Scholar

[10]

P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions,, Communications on Pure and Applied Mathematics, 37 (1984), 511.  doi: 10.1002/cpa.3160370408.  Google Scholar

[11]

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.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[12]

H. Meng and T. Siu, On optimal reinsurance, dividend and reinvestment strategies,, Economic Modelling, 28 (2011), 211.  doi: 10.1016/j.econmod.2010.09.009.  Google Scholar

[13]

M. Ohnishi and M. Tsujimura, An impulse control of a geometric Brownian motion with quadratic costs,, European Journal of Operational Research, 168 (2006), 311.  doi: 10.1016/j.ejor.2004.07.006.  Google Scholar

[14]

J. Paulsen, Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs,, Advances in Applied Probability, 39 (2007), 669.  doi: 10.1239/aap/1189518633.  Google Scholar

[15]

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.  doi: 10.1137/070691632.  Google Scholar

[16]

S. P. Sethi and M. Taksar, Optimal financing of a corporation subject to random retures,, Mathematical Finance, 12 (2002), 155.  doi: 10.1111/1467-9965.t01-2-02002.  Google Scholar

[17]

S. E. Shreve, J. P. Lehoczky and D. P. Gaver, Optimal consumption for general diffusions with absorbing and reflecting barriers,, SIAM Journal on Control and Optimization, 22 (1984), 55.  doi: 10.1137/0322005.  Google Scholar

[18]

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.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[19]

D. Yao, H. Yang and R. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle,, Economic Modelling, 37 (2014), 53.  doi: 10.1016/j.econmod.2013.10.026.  Google Scholar

[1]

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
[2]

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
[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]

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
[5]

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
[6]

Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2020008
[7]

Min Dai, Zhou Yang. A note on finite horizon optimal investment and consumption with transaction costs. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1445-1454. doi: 10.3934/dcdsb.2016005
[8]

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
[9]

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
[10]

Qianru li, Weida chen, Yongming zhang. Optimal production and emission reduction policies for a remanufacturing firm considering deferred payment strategy. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020078
[11]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[12]

Francesca Biagini, Thilo Meyer-Brandis, Bernt Øksendal, Krzysztof Paczka. Optimal control with delayed information flow of systems driven by G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 8-. doi: 10.1186/s41546-018-0033-z
[13]

Zhongbao Zhou, Ximei Zeng, Helu Xiao, Tiantian Ren, Wenbin Liu. Multiperiod portfolio optimization for asset-liability management with quadratic transaction costs. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1493-1515. doi: 10.3934/jimo.2018106
[14]

Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic & Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001
[15]

Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169
[16]

Yiling Chen, Baojun Bian. optimal investment and dividend policy in an insurance company: A varied bound for dividend rates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5083-5105. doi: 10.3934/dcdsb.2019044
[17]

Ka Wo Lau, Yue Kuen Kwok. Optimal execution strategy of liquidation. Journal of Industrial & Management Optimization, 2006, 2 (2) : 135-144. doi: 10.3934/jimo.2006.2.135
[18]

Zhen Wang, Sanyang Liu. Multi-period mean-variance portfolio selection with fixed and proportional transaction costs. Journal of Industrial & Management Optimization, 2013, 9 (3) : 643-656. doi: 10.3934/jimo.2013.9.643
[19]

Wen Li, Song Wang. Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. Journal of Industrial & Management Optimization, 2013, 9 (2) : 365-389. doi: 10.3934/jimo.2013.9.365
[20]

Donny Citra Lesmana, Song Wang. A numerical scheme for pricing American options with transaction costs under a jump diffusion process. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1793-1813. doi: 10.3934/jimo.2017019

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences