[Math] Ito’s formula for multivariable Ito integral

Question

I'm having trouble finding something that I think should exist, which is an integral formula of the multivariable Ito lemma. Simply put, suppose I have a function $f$ of two stochastic processes, $X_1(s)$ and $X_1(s)$ that are as well behaved as you would like and I also happen to know the quadratic variations $[X_1]_t$ and $[X_1,X_2]_t$. I would like to apply Ito's lemma to the integral
$$
\int_0^t f(X_1(s),X_2(s))dX_1(s).
$$
Looking at the differential form of the multivariable Ito lemma, it seems like this equals something like
$$
\int_0^{X_1(t)}f(\xi,X_2(s))d\xi – \int_0^{X_1(0)}f(\xi,X_2(s))d\xi \\- \frac{1}{2}\int_0^t\frac{\partial}{\partial X_1}f(X_1(s),X_2(s))d[X_1]_s-\frac{1}{2}\int_0^t\frac{\partial}{\partial X_2}f(X_1(s),X_2(s))d[X_1,X_2]_s.
$$
But this seems kind of weird because the answer still depends on the time $s$ that I look at $X_2$—certainly this can't be. So what's going wrong here?

Answer 1

So what's going wrong here?

That the $\mathrm dX_2$ and $\mathrm d[X_1,X_2]$ terms are missing.

More in details, let $F$ such that $\partial_1F=f$, then Itô's formula reads $$ \begin{align} \mathrm dF(X_1(t),X_2(t)) &=f(X_1(t),X_2(t))\mathrm dX_1(t)+\partial_2F(X_1(t),X_2(t))\mathrm dX_2(t)+ \\ &\qquad +\tfrac12\partial_1f(X_1(t),X_2(t))\mathrm d[X_1](t)+\partial_2f(X_1(t),X_2(t))\mathrm d[X_1,X_2](t)+ \\ &\qquad+\tfrac12\partial^2_{22}F(X_1(t),X_2(t))\mathrm d[X_2](t). \end{align} $$