Finite Horizon Optimal Investment and Consumption with CARA Utility and Proportional Transaction costs∗

We are concerned with the optimal investment and consumption in a continuous-time setting for a constant absolute risk aversion (CARA) investor who faces proportional transaction costs and finite horizon. This is a singular stochastic control problem and the associated value function is governed by a variational inequality equation with gradient constraints. In terms of a novel approach developed by Dai and Yi [J. Differential Equations, 246 (2009), pp. 1445-1469], we provide a comprehensive theoretical analysis on the optimal investment and consumption strategy. It turns out that the optimal investment strategy is characterized by the resulting free boundaries which exhibit similar behaviors as in Dai et al. [SIAM J. Control Optim., 48 (2009), pp. 1134-1154] where the constant relative risk aversion utility is considered. Moreover, we show that in order to maintain optimal consumption, the CARA investor needs to invest more in risky asset. In addition, with the CARA utility, the optimal consumption may be negative. To ensure positive consumption for any positive liquidated wealth in the presence of transaction costs, the sufficient and necessary condition is that the discount rate is not less than the risk-free rate.