Ito Formula

$F(X_t^1,X_t^2,...,X_t^d)=F(X_0^1,X_0^2,...,X_0^d)+{\sum }_{i=1}^d{\int }_0^t\frac{\partial }{\partial x^i}F(X_s^1,X_s^2,...,X_s^d)d{X}_s^i+\frac{1}{2}{\sum }_{1\le i,j\le d}{\int }_0^t\frac{{\partial }^2}{\partial x^i\partial x^j}F(X_s^1,X_s^2,...,X_s^d)d{⟨X^i,X^j⟩}_s$

• $X_t^i$ are $d$ distinct continuous Semimartingale indexed by $i$.
• $F:{\mathbb{R}}^d\to \mathbb{R}\in C^2$

For $d=1$, we get the simpler calculation

$F(X_t)=F(X_0)+{\int }_0^t{F}^{\prime }(X_s)d{X}_s+\frac{1}{2}{\int }_0^t{F}^{\prime \prime }(X_s)d{⟨X⟩}_s$

Proof

See Theorem 9.3, in page 60, of https://courses.maths.ox.ac.uk/pluginfile.php/103016/mod_resource/content/2/LectureNotes24.pdf