Finite Horizon Optimal Investment and Consumption with Transaction Costs

Abstract

This paper concerns continuous-time optimal investment and the consumption decision of a constant relative risk aversion (CRRA) investor who faces proportional transaction costs and a finite time horizon. In the no-consumption case, it has been studied by Liu and Loewenstein [Review of Financial Studies, 15 (2002), pp. 805–835] and Dai and Yi [J. Differential Equations, 246 (2009), pp. 1445–1469]. Mathematically, it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraints. The problem gives rise to two free boundaries which stand for the optimal buying and selling strategies, respectively. We present an analytical approach to analyze the behaviors of free boundaries. The regularity of the value function is studied as well. Our approach is essentially based on the connection between singular control and optimal stopping, which is first revealed in the present problem.