  We consider a class of time-inhomogeneous optimal stopping problems and we
provide sufficient conditions on the data of the problem that guarantee
monotonicity of the optimal stopping boundary. In our setting,
time-inhomogeneity stems not only from the reward function but, in particular,
from the time dependence of the drift coefficient of the one-dimensional
stochastic differential equation (SDE) which drives the stopping problem. In
order to obtain our results, we mostly employ probabilistic arguments: we use a
comparison principle between solutions of the SDE computed at different
starting times, and martingale methods of optimal stopping theory. We also show
a variant of the main theorem, which weakens one of the assumptions and
additionally relies on the renowned connection between optimal stopping and
free-boundary problems.
