Wat is Quadratic variation?

The quadratic variation of a process $X$ on $[0, t]$ is the limit in probability $[X]_t = \lim\sum_k (X_{t_{k+1}} - X_{t_k})^2$ over partitions $0 = t_0 < \dots < t_n = t$ whose mesh $\max_k(t_{k+1} - t_k)$ tends to zero, when it exists.