The Ito's formula stated in my book is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and $(X_t)_{t \geq0}$ is a $d-$ dimensional predictable process.
I am wondering whether the function can be random, e.g. for any stopping time $\tau$, can we apply Ito's formula to the process $$ \{ F(t,X_t) \, \mathbf{1}_{ \{ \tau \leq t \} } \}_{t \geq 0} \quad \quad ?$$