# Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $$B$$ over the interval $$[0,t]$$ is defined as the limit in probability of any sequence of partitions $$\Pi_n([0,t])=\{0=t^n_0<\cdots of the interval $$[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.

• In addition to the answer below, these MIT lecture notes here are also pretty decent. Jan 14, 2021 at 12:35

• @TheBridge : Yes, I agree. Also if you are interested in a more precise result for BM, i.e. the "largest" $\varphi$ s.t. BM has finite $\varphi$-variation a.s., you can find in the paper "Exact asymptotic estimates of Brownian path variation",SJ Taylor, Duke Math. J. Volume 39, Number 2 (1972), 219-241. Dec 21, 2011 at 13:16