  We propose an alternative approach for solving a number of well-studied
optimal stopping problems for L\'evy processes. Instead of the usual method of
guess-and-verify based on martingale properties of the value function, we
suggest a more direct method by showing that the general theory of optimal
stopping for strong Markov processes together with some elementary observations
imply that the stopping set must be of a certain form for the optimal stopping
problems we consider. The independence of increments and the strong Markov
property of L\'evy processes then allow us to use straightforward optimisation
over a real-valued parameter to determine this stopping set. We illustrate this
approach by applying it to the McKean optimal stopping problem (American put),
the Novikov--Shiryaev optimal stopping problem and the Shepp--Shiryaev optimal
stopping problem (Russian option).
