  Let $G$ be a semimartingale, and $S$ its Snell envelope. Under the assumption
that $G\in\mathcal{H}^1$, we show that the finite-variation part of $S$ is
absolutely continuous with respect to the decreasing part of the
finite-variation part of $G$. In the Markovian setting, this enables us to
identify sufficient conditions for the value function of the optimal stopping
problem to belong to the domain of the extended (martingale) generator of the
underlying Markov process. We then show that the \textit{dual} of the optimal
stopping problem is a stochastic control problem for a controlled Markov
process, and the optimal control is characterised by a function belonging to
the domain of the martingale generator. Finally, we give an application to the
smooth pasting condition.
