I am reviewing this paper and I'm on page 3 of the document, and I'm having trouble with the proof of uniqueness.
First off, the version of Ito's lemma I've learned is: if $X_{t}$ is an Ito process (i.e., $X_{t} = X_{0} + \int \limits_{0}^{t} b(s, \omega) \,ds + \int \limits_{0}^{t} a(s,\omega) \, dB_{s}$) and if $f$ is twice differentiable, then $$f(X_{t}) = f(X_{0}) + \int \limits_{0}^{t} f'(X_{s}) b(s, \omega) \,ds + a(s,\omega) \,dB_{s} + \frac{1}{2}\int \limits_{0}^{t} f''(X_{s}) a^{2}(s, \omega) \,ds. $$
My first question is: How would Ito's formula be different considering the paper is dealing with Backward Stochastic Differential Equations where the limits of integration are from $t$ to $1$?
Now, the authors suppose that there are two solutions $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$. They apply Ito's formula to $(x_{1} - x_{2})^{2}$. I'm assuming here that the $f$ from the Ito's formula I have above is $f(x) = x^{2}$, which is $C^{2}$.
My next questions are: I don't really understand how they applied Ito's formula to get the resulting equation. It seems like the integrand in each integral is a vector with one of the components the function $x_{1} - x_{2}$. Where did this come from?
Regarding the first question: The $0$ and $t$ in the typical statement of Ito's Lemma aren't required. In general you can say: $$f(X_{v}) = f(X_{u}) + \int \limits_{u}^{v} f'(X_{s}) b(s, \omega) \,ds + a(s,\omega) \,dB_{s} + \frac{1}{2}\int \limits_{u}^{v} f''(X_{s}) a^{2}(s, \omega) \,ds. $$ Then set $u = 0, v = t$ for the typical statement, and $u = t, v = 1$ for the version used in the paper.
Regarding the second question: If you reexamine the definition of the processes in the proposition you will see their range is $\mathbb{R}^d$, so they are vector valued. You need to use the Multidimensional Ito's Lemma which you can find on that page.