Abstract: 
We study a dynamic optimization problem with both stochastic control and optimal stopping in a finite time horizon. The problem can be converted into an equivalent free boundary problem of a fully nonlinear PDE. We give rigorous analysis of the dual control method for this problem with a class of utility functions, including power and nonHARA utilities. The dual problem is still a free boundary problem, but the governing PDE is linear. We analyse the asymptotic properties of the free boundary of the dual problem, construct a global closedform approximation of the free boundary for the dual problem, and obtain the approximate formula for the dual value function, which in turn gives the approximate formulas for the primal value function, optimal control and optimal exercise boundary for the optimal investment stopping problem. Numerical examples show that the formulas are accurate and fast. 
