Theorem (Quadratic variation)

Let $M$ be a Continuous local martingale. Let $t\ge 0$, and let $(\pi_{t}^{n})$ be a sequence of subdivisions of $[0,t]$ with mesh $|\pi_{t}^{n}|\to 0$ as $n\to \infty$. For $n\in\mathbb{N}$, let $$S_{t}^{n}=\sum_{i=0}^{N-1}(M_{t_{i+1}}-M_{t_{i}})^{2}$$ where $\pi_{t}^{n}=(t_{0},\dots,t_{N})$. Then:

1. If $M$ is bounded, then $(S_{t}^{n})_{n\in\mathbb{N}}$ converges in $L^{2}$ to $$[M]_{t}=M_{t}^{2}-M_{0}^{2}-2\int\limits \mathbb{1}_{[0,t]}M \, dM$$
2. $(S_{t}^{n})_{n\in\mathbb{N}}$ converges in probability to $[M]_{t}$

$([M]_{t})_{t\ge 0}$ is called the quadratic variation process of $M$.