A stochastic portfolio optimization model on an infinite time horizon is considered. The goal is to choose the optimal strategy for investment and consumption rate to maximize the total expected, discounted utility. A dynamic programming principle is used to derive the dynamics programming equation for the value function. The sub/super solution method is used to get the existence of the solution. The solution is then used to obtain the optimal control policies for investment and consumption rate.
This is a generalization of the classical Merton's problem. The background of Merton's problem will be given in the beginning of the talk. I will then talk about some technical details about the problem I dealt with.