  A \emph{new} notion of equilibrium, which we call \emph{strong equilibrium},
is introduced for time-inconsistent stopping problems in continuous time.
Compared to the existing notions introduced in ArXiv: 1502.03998 and ArXiv:
1709.05181, which in this paper are called \emph{mild equilibrium} and
\emph{weak equilibrium} respectively, a strong equilibrium captures the idea of
subgame perfect Nash equilibrium more accurately. When the state process is a
continuous-time Markov chain and the discount function is log sub-additive, we
show that an optimal mild equilibrium is always a strong equilibrium. Moreover,
we provide a new iteration method that can directly construct an optimal mild
equilibrium and thus also prove its existence.
