  In this paper we study a utility maximization problem with both optimal
control and optimal stopping in a finite time horizon. The value function can
be characterized by a variational equation that involves a free boundary
problem of a fully nonlinear partial differential equation. Using the dual
control method, we derive the asymptotic properties of the dual value function
and the associated dual free boundary for a class of utility functions,
including power and non-HARA utilities. We construct a global closed-form
approximation to the dual free boundary, which greatly reduces the
computational cost. Using the duality relation, we find the approximate
formulas for the optimal value function, trading strategy, and exercise
boundary for the optimal investment stopping problem. Numerical examples show
the approximation is robust, accurate and fast.
