  This paper studies an optimal stopping problem for L\'evy processes. We give
a justification of the form of the Snell envelope using standard results of
optimal stopping. We also justify the convexity of the value function, and
without a priori restriction to a particular class of stopping times, we deduce
that the smallest optimal stopping time is necessarily a hitting time. We
propose a method which allows to obtain the optimal threshold. Moreover this
method allows to avoid long calculations of the integro-differential
operatorused in the usual proofs.
