Almost strong equilibria for time-inconsistent stopping problems under finite horizon in continuous time

We consider time-inconsistent stopping problems for a continuous-time Markov chain under finite time horizon with non-exponential discounting. We provide an example indicating that strong equilibria may not exist in general. As a result, we propose a notion of equilibrium called almost strong equilibrium (ASE), which is a weak equilibrium and satisfies the condition of strong equilibria except at the boundary points of the associated stopping region. We provide an iteration procedure and show that this procedure leads to an ASE. Moreover, we prove that this ASE is the unique ASE among all regular stopping policies under finite horizon $T<\infty$. In contrast, we show that strong equilibria (and thus ASE) exist and may not be unique for the infinite horizon case $T=\infty$. Furthermore, we show that the limit of the finite-horizon ASE as $T\to \infty$ is a weak equilibrium for the infinite-horizon problem, and may not be a strong equilibrium or ASE.