Optimal investment consumption model with vasicek interest rate

The optimal investment consumption problem for a single riskless bond, a zero-coupon bond and a risky stock modeled by the Vasicek interest process has been established. The investment objective is maximizing the utility of his consumption and terminal wealth. By the stochastic dynamic programming principle, the HJB equation for the optimal solution is given. In the case of constant relative risk aversion utility, the analytic optimal trading strategies are derived. The results show that the optimal proportion allocated in the stock is a constant fraction, but the optimal proportion in the zero-coupon bond is time-variant. The optimal consumption rate is in a feedback form of the wealth and depends on the stochastic interest rate. A numerical example illustrating the results is presented.