An Optimal investment and consumption problem with the special coefficient logarithm utility

This paper concerns with portfolio problems where an investor can invest in a savings account, stocks and bonds and tries to maximize her utility from terminal wealth and her expected utility of intermediate consumption. We apply stochastic control method to solve the problem. By using logarithmic utility of special constant, we firstly set up the HJB-equation satisfied by the value function of the portfolio problem. Then we obtain its explicit solution. Furthermore, optimal investment and consumption policies are presented. Therefore, we solve this problem completely.