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

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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

Abstract
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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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