Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the interval $[0,t]$ such that $\lim_{n\to \infty}\max_{i=1,\cdots,k(n)}|t^n_i-t^n_{i-1}|=0$, of the functional
$$V([0,t],\Pi_n)(B_.)=\sum_{i=1}^{k(n)}(B_{t_{i-1}}-B_{t_i})^2.$$
And for any such sequence of partition we have then $[B]_t=P-\lim_{n\to \infty} V([0,t],\Pi_n)(B_.)=t$.
Nevertheless when you take the sup over all finite partitions of $[0,t]$ then it is a known fact that almost surely $\sup_{\Pi\in \mathrm{partition}([0,t])} V([0,t],\Pi)(B_.)=+\infty$.
I have never been able to derive this fact properly and in every details.
I'd be really gratefull if anyone could take the time to provide a detailed proof of this fact.
Best Answer
You can find a short proof of this fact (actually in the more general case of Fractional Brownian Motion) in the paper :
M. Prattelli : A remark on the 1/H-variation of the Fractional Brownian Motion. Probability Seminar Vol. XLIII (pdf)