Before revision:  The time-homogeneous diffusion process in this paper includes the geometric Brownian motion process , the CEV process, the mean-reverting Gaussian process or the Vasicek model, and the mean-reverting square root process or the CIR model. The contributions are to provide rigorous proofs of following facts. The value of an American-style put option increases with an increase in the time-to-maturity and decreases with an increase in the underlying asset index. The early exercise boundary is a strictly decreasing convex functionof the time-to-maturity under the given conditions .
After revision:  The time-homogeneous diffusion process in this paper includes the geometric Brownian motion process . The contribution is to propose a condition for that the early exercise boundary is a strictly decreasing convex functionof the time-to-maturity under the given conditions .