  This article treats both discrete time and continuous time stopping problems
for general Markov processes on the real line with general linear costs. Using
an auxiliary function of maximum representation type, conditions are given to
guarantee the optimal stopping time to be of threshold type. The optimal
threshold is then characterized as the root of that function. For random walks
our results condense in the fact that all combinations of concave increasing
pay-off functions and convex cost functions lead to a one-sided solution. For
L\'evy processes an explicit way to obtain the auxiliary function and the
threshold is given by use of the ladder height processes. Lastly, the
connection from discrete and continuous problem and possible approximation of
the latter one via the former one is discussed.
