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.
fhyi@scnu.edu.cn
³ Department of Economics, University of Kansas, Lawrence KS66045, USA.

Received: 23 October 2015
Revised: 16 April 2016
Accepted: 22 June 2016

Abstract

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.