Theorem (Mutual variation of Itô Stochastic Integrals)

Let $M,N$ be continuous local martingales. Let $X\in\Lambda^{2}(\mathscr{P},M)$, $Y\in\Lambda^{2}(\mathscr{P},N)$. We denote $X\cdot M$ as $$X\cdot M=\left( \int\limits \mathbb{1}_{[0,t]}X \, dM \right)_{t\ge 0}$$Then $$[X\cdot M,Y\cdot N]_{t}=\int\limits _{0}^{t}X_{s}Y_{s} \, d[M,N]_{s}\quad\forall t\ge 0$$In particular: $$[X\cdot M]_{t}=\int\limits _{0}^{t}X_{s}^{2} \, d[M]_{s}$$