  We consider the problem of optimally stopping a general one-dimensional
stochastic differential equation (SDE) with generalised drift over an infinite
time horizon. First, we derive a complete characterisation of the solution to
this problem in terms of variational inequalities. In particular, we prove that
the problem's value function is the difference of two convex functions and
satisfies an appropriate variational inequality in the sense of distributions.
We also establish a verification theorem that is the strongest one possible
because it involves only the optimal stopping problem's data. Next, we derive
the complete explicit solution to the problem that arises when the state
process is a skew geometric Brownian motion and the reward function is the one
of a financial call option. In this case, we show that the optimal stopping
strategy can take several qualitatively different forms, depending on parameter
values. Furthermore, the explicit solution to this special case reveals that
the so-called "principle of smooth fit" does not hold in general for this type
of optimal stopping problems in standard senses that this can be formulated.
