  We derive a change of variable formula for $C^1$ functions
$U:\mathbb{R}_+\times\mathbb{R}^m\to\mathbb{R}$ whose second order spatial
derivatives may explode and not be integrable in the neighbourhood of a surface
$b:\mathbb{R}_+\times\mathbb{R}^{m-1}\to \mathbb{R}$ that splits the state
space into two sets $\mathcal{C}$ and $\mathcal{D}$. The formula is tailored
for applications in problems of optimal stopping where it is generally very
hard to control the second 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.
