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.

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

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)

  • $\begingroup$ @ pgassiat : Thank's this looks promising let me take a look at it. $\endgroup$
    – TheBridge
    Dec 20 '11 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 '11 at 17:17
  • $\begingroup$ @The Bridge : You're welcome. $\endgroup$
    – pgassiat
    Dec 20 '11 at 18:20
  • 2
    $\begingroup$ IMO, this should be added to every stochastic calculus book when talking about quadratic variation of Brownian Motion. $\endgroup$
    – TheBridge
    Dec 21 '11 at 7:38
  • 1
    $\begingroup$ @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 '11 at 13:16

The book Brownian Motion - An Introduction to Stochastic Processes by René Schilling & Lothar Partzsch contains a (detailed) proof of this fact.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.