Volume 23, Number 4, October-December 2017
|Page(s)||1601 - 1615|
|Published online||21 September 2017|
Investment and consumption problem in finite time with consumption constraint∗
1 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China.
2 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, P.R. China.
3 Department of Economics, University of Kansas, Lawrence KS66045, USA.
Received: 23 October 2015
Revised: 16 April 2016
Accepted: 22 June 2016
In this paper, we consider an investment-consumption problem where the consumption is subject to an upper limit. This upper limit on consumption may reflect the following fact. Investors may have to finance their consumption first by using credits then pay the balance by cashing out part of their portfolio in the stock market. Credit companies set up an upper limit for the credit, thus imposing an upper bound for consumption. We also set up our model in finite horizon, which makes the problem much harder due to the loss of stationary when T < ∞. We prove that the above described problem is equivalent to a free boundary problem of nonlinear parabolic equations. We aim to characterize explicitly the free boundary by applying a dual transformation technique to convert the original nonlinear parabolic equation to a linear differential equation. This trick allows us to characterize explicitly the free boundary and the optimal consumption strategy. We also prove that the regularity of the value function, which is critical for the application of Ito formula.
Mathematics Subject Classification: 35R35 / 91B28 / 93E20
Key words: Optimal investment-consumption model / free boundary problem / stochastic control in finance / consumption constraint
The project is supported by NNSF of China (No. 11271143, No. 11371155, Nos. 11471276 and 71471045), NSF of Guangdong Province of China (Nos. 2015A030313574 and 2016A030313448), The Humanities and Social Science Research Foundation of the National Ministry of Education of China (No. 15YJAZH051) and University Special Research Fund for Ph.D. Program of China (20124407110001).
© EDP Sciences, SMAI 2017
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