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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.

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

2 Answers 2

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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)

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  • $\begingroup$ @ pgassiat : Thank's this looks promising let me take a look at it. $\endgroup$
    – TheBridge
    Dec 20, 2011 at 16:14
  • $\begingroup$ Excellent proof !!! And note that for BM lemma 1 simplifies further as it is only Strong Law of Large Numbers (Brownian increments are i.i.d. in this context). Really thank you so much, I have been looking for such a detailed proof for years !!! $\endgroup$
    – TheBridge
    Dec 20, 2011 at 17:17
  • $\begingroup$ @The Bridge : You're welcome. $\endgroup$
    – pgassiat
    Dec 20, 2011 at 18:20
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    $\begingroup$ IMO, this should be added to every stochastic calculus book when talking about quadratic variation of Brownian Motion. $\endgroup$
    – TheBridge
    Dec 21, 2011 at 7:38
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    $\begingroup$ @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. $\endgroup$
    – pgassiat
    Dec 21, 2011 at 13:16
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The book Brownian Motion - An Introduction to Stochastic Processes by René Schilling & Lothar Partzsch contains a (detailed) proof of this fact.

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