Let $X_t$ be an Itô process satisfying $$ dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dB_t, $$ $t\in I$, being $I$ an interval in $(0,\infty)$ and $B_t$ a standard Brownian motion. I studied that Itô's formula can be applied to $f(t,X_t)$ when $f\in C^{1,2}(I\times\mathbb{R})$. Let $\mathcal{D}$ be an open set of $\mathbb{R}$ such that $X(t)\in \mathcal{D}$ for all $t\in I$. My question is whether the assumption $f\in C^{1,2}(I\times\mathcal{D})$ is sufficient to apply Itô's formula.
Motivation: Consider the lognormal model $dS_t=\mu S_t\,dt+\sigma S_t\,dB_t$ for the asset price $S_t>0$. To find the solution, Itô's formula is applied to $f(x)=\log x$. But $f$ is $C^2$ on $(0,\infty)$, not on the whole $\mathbb{R}$.
Yes, it's enough to assume that $f \in C^{1,2}(I \times D)$.
The reason is, essentially, that Itô's formula can be applied to stopped processes. If we define $$\tau_D := \inf\{t>0; X_t \notin D\}$$
then $\tau_D$ is a stopping time (with respect to the filtration $(\mathcal{F}_{t+})_{t \geq 0})$, and therefore Itô's formula gives
$$\begin{align*} f(t,X_{t \wedge \tau_D})-f(0,X_0) &= \int_0^{t \wedge \tau_D} \partial_x f(s,X_s) \, dX_s \\ &\quad + \int_0^{t \wedge \tau_D} \left( \frac{\sigma^2(s,X_s)}{2} \partial_x^2 f(s,X_s) + \partial_t f(s,X_s) \right) \, ds.\end{align*}$$
Since $(X_t)_{t \geq 0}$ takes only values in $D$, we have $\tau_D = \infty$ almost surely, and this gives the desired identity.