  For a type of employee stock option (ESO) and an American put option with a
barrier, we obtain closed-form formulae for the value functions and provide a
complete characterization for optimal stopping/continuation regions. Some
comparison principles for the critical levels and the value functions are
given. This work is inspired by the characterization of the value functions for
general one-dimensional regular diffusion processes developed in \cite{DK03} by
Dayanik and Karatzas.
