Given , Continuous Local Martingales then the quadratic covariation is the Stochastic Process defined by Misplaced &&= \lim_{n \to \infty} \sum_{i=0}^{n-1} (M_{t_{i+1}} - M_{t_i})(N_{t_{i+1}} - N_{t_i}) \space\space\space\space \text{in} \space \mathbb{P}\end{aligned}$$ such that 1. $\left\langle M, N \right\rangle_t$ is a [[Finite Variation Process]] 2. $\left\langle M, N \right\rangle_t$ is unique up to indistinguishability 3. $M_t N_t - \left\langle M, N \right\rangle_t$ is a CLM $\left\langle M \right\rangle_t$ has the following properties 1. If $M \perp\!\!\!\perp N$ $$\left\langle M, N \right\rangle_t = 0$$ 2. $T$ a stopping time $$\left\langle M^T, N^T \right\rangle_t = \left\langle M^T, N \right\rangle_t = \left\langle M, N^T \right\rangle_t = \left\langle M, N \right\rangle_{t \wedge T}$$ 3. If $M$, $N$ are true [[Martingale]]s bounded in $L^2$, i.e. $\sup_{t \geq 0} \mathbb{E}[M_t^2] < \infty$ and $\sup_{t \geq 0} \mathbb{E}[N_t^2] < \infty$ $$M_t N_t - \left\langle M, N \right\rangle_t \space \text{is a u.i. Martingale and} \lim_{t \to \infty} \left\langle M, N \right\rangle_t \space \text{exists a.s}$$