Theorem (Itô’s Formula)

Let $(M_{t})_{t\ge 0}$ be a continuous local martingale, let $(V_{t})_{t\ge 0}$ be a continuous process of Finite Variation (i.e. locally bounded). Let $f\in C^{0}(\mathbb{R}^{2}; \mathbb{R})$ with $\frac{ \partial f }{ \partial x },\frac{ \partial f }{ \partial y },\frac{ \partial ^{2}f }{ \partial x^{2} }$ $C^{0}$ on $\mathbb{R}^{2}$. Then we have a.s. $\forall t\ge 0$: $$\begin{align*} &f(M_{t},V_{t})\\&= f(M_{0},V_{0})+\int\limits \mathbb{1}_{[0,t]}\frac{ \partial f }{ \partial x } (M,V) \, dM +\int\limits _{[0,t]}\frac{ \partial f }{ \partial y } (M_{s},V_{s}) \, dV_{s} + \frac{1}{2}\int\limits _{[0,t]} \frac{ \partial ^{2}f }{ \partial x^{2} } (M_{s},V_{s}) \, d[M]_{s} \end{align*}$$