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September  2017, 7(3): 465-491. doi: 10.3934/mcrf.2017017

Investment and consumption in regime-switching models with proportional transaction costs and log utility

1. 

School of Economics and Management, China Jiliang University, 258 Xueyuan Road, Hangzhou, Zhejiang 310018, China

2. 

Department of Mathematics, University of Dayton, 300 College Park, Dayton, OH 45469-2316, USA

* Corresponding author: Ruihua Liu

Received  August 2016 Revised  October 2016 Published  July 2017

A continuous-time and infinite-horizon optimal investment and consumption model with proportional transaction costs and regime-switching was considered in Liu [4]. A power utility function was specifically studied in [4]. This paper considers the case of log utility. Using a combination of viscosity solution to the Hamilton-Jacobi-Bellman (HJB) equation and convex analysis of the value function, we are able to derive the characterizations of the buy, sell and no-transaction regions that are regime-dependent. The results generalize Shreve and Soner [6] that deals with the same problem but without regime-switching.

Citation: Jiapeng Liu, Ruihua Liu, Dan Ren. Investment and consumption in regime-switching models with proportional transaction costs and log utility. Mathematical Control and Related Fields, 2017, 7 (3) : 465-491. doi: 10.3934/mcrf.2017017
References:
[1]

J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.

[2]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[3]

M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676-713.  doi: 10.1287/moor.15.4.676.

[4]

R. H. Liu, Optimal investment and consumption with proportional transaction costs in regime-switching model, J Optim Theory Appl, 163 (2014), 614-641.  doi: 10.1007/s10957-013-0445-y.

[5]

P. E. Protter, Stochastic Integration and Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 2005. doi: 10.1007/978-3-662-02619-9.

[6]

S. E. Shreve and H. M. Soner, Optimal investment and consumption with transaction costs, The Annals of Applied Probability, 4 (1994), 609-692.  doi: 10.1214/aoap/1177004966.

[7]

R. TaoZ. Wu and Q. Zhang, Optimal switching under a regime-switching model with two-time-scale Markov chains, Multiscale Model. Simul., 13 (2015), 99-131.  doi: 10.1137/130938967.

[8]

T. Zariphopoulou, Investment-consumption models with transaction fees and Markov-chain parameters, SIAM J. Control and Optimization, 30 (1992), 613-636.  doi: 10.1137/0330035.

show all references

References:
[1]

J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.

[2]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[3]

M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676-713.  doi: 10.1287/moor.15.4.676.

[4]

R. H. Liu, Optimal investment and consumption with proportional transaction costs in regime-switching model, J Optim Theory Appl, 163 (2014), 614-641.  doi: 10.1007/s10957-013-0445-y.

[5]

P. E. Protter, Stochastic Integration and Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 2005. doi: 10.1007/978-3-662-02619-9.

[6]

S. E. Shreve and H. M. Soner, Optimal investment and consumption with transaction costs, The Annals of Applied Probability, 4 (1994), 609-692.  doi: 10.1214/aoap/1177004966.

[7]

R. TaoZ. Wu and Q. Zhang, Optimal switching under a regime-switching model with two-time-scale Markov chains, Multiscale Model. Simul., 13 (2015), 99-131.  doi: 10.1137/130938967.

[8]

T. Zariphopoulou, Investment-consumption models with transaction fees and Markov-chain parameters, SIAM J. Control and Optimization, 30 (1992), 613-636.  doi: 10.1137/0330035.

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