If $(X_t)$ is a continuous $d$-dimensional semimartingale and $f$ is $C^2$, we have Itô's lemma
$$f\left(X_{t}\right)=f\left(X_{0}\right)+\sum_{i=1}^{d} \int_{0}^{t} f_{i}\left(X_{s}\right) d X_{s}^{i}+\frac{1}{2} \sum_{i, j=1}^{d} \int_{0}^{t} f_{i, j}\left(X_{s}\right) d\left[X^{i}, X^{j}\right]_{s}$$
If we have $f(t,X_t)$ instead, I am tempted to say that the formula becomes
$$f\left(t,X_{t}\right)=f\left(0,X_{0}\right)+ \int_{0}^{t} \frac{\partial f}{\partial t}\left(s,X_{s}\right) d s+\sum_{i=1}^{d} \int_{0}^{t} f_{i}\left(s,X_{s}\right) d X_{s}^{i}+\frac{1}{2} \sum_{i, j=1}^{d} \int_{0}^{t} f_{i, j}\left(s,X_{s}\right) d\left[X^{i}, X^{j}\right]_{s}$$
Is this true ? Do you have a reference ?