Before revision:  Furthermore, we provide a rigorous veri?cation for that the free boundary for this problem is concave. As an application to the American option pricing problem, our results imply that the early exercise boundary of an American call is a strictly decreasing concave function. This result provides a useful information to obtain an asymptotic formula for the early exercise boundary .
After revision:  Furthermore, we provide a rigorous veri?cation for that the free boundary for this problem is convex .