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.
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