  We derive a change of variable formula for $C^1$ functions
$U:\R_+\times\R^m\to\R$ whose second order spatial derivatives may explode and
not be integrable in the neighbourhood of a surface $b:\R_+\times\R^{m-1}\to
\R$ that splits the state space into two sets $\cC$ and $\cD$. The formula is
tailored for applications in problems of optimal stopping where it is generally
very hard to control the second order derivatives of the value function near
the optimal stopping boundary. Differently to other existing papers on similar
topics we only require that the surface $b$ be monotonic in each variable and
we formally obtain the same expression as the classical It\^o's formula.
