Say I have the following level set: $c = f(x,y)$, where $x$ is a stochastic variable, and $y$ is purely deterministic, and of course, $c \in \mathbb{R}$. Does applying Ito's lemma to $c$ make sense? I.e.,
$dc = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{1}{2}\frac{\partial^2 f}{\partial x^2} (dx)^2 + \frac{1}{2} \frac{\partial^2 f}{\partial y^2} (dy)^2$.
Is it appropriate then, to write $dc = 0$?
Thanks!