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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 & 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. Google Scholar [2] M. G. Crandall, H. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] R. Tao, Z. 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. Google Scholar [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. Google Scholar

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. Google Scholar [2] M. G. Crandall, H. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] R. Tao, Z. 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. Google Scholar [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. Google Scholar
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