  In the literature on optimal stopping, the problem of maximizing the expected
discounted reward over all stopping times has been explicitly solved for some
special reward functions (including $(x^+)^{\nu}$, $(e^x-K)^+$, $(K-e^{-x})^+$,
$x\in\mathbb{R}$, $\nu\in(0,\infty)$ and $K>0$) under general random walks in
discrete time and L\'evy processes in continuous time (subject to mild
integrability conditions). All of such reward functions are continuous,
increasing and logconcave while the corresponding optimal stopping times are of
threshold type (i.e. the solutions are one-sided). In this paper, we show that
all optimal stopping problems with increasing, logconcave and right-continuous
reward functions admit one-sided solutions for general random walks and L\'evy
processes. We also investigate in detail the principle of smooth fit for L\'evy
processes when the reward function is increasing and logconcave.
