  We obtain a probabilistic proof of the local Lipschitz continuity for the
optimal stopping boundary of a class of problems with state space
$[0,T]\times\mathbb{R}^d$, $d\ge 1$. To the best of our knowledge this is the
only existing proof that relies exclusively upon stochastic calculus, all the
other proofs making use of PDE techniques and integral equations. Thanks to our
approach we obtain our result for a class of diffusions whose associated second
order differential operator is not necessarily uniformly elliptic. The latter
condition is normally assumed in the related PDE literature.
