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Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval is defined by $$V_{p}(x, [a, b]) = \lim_{|\Pi| \to 0} \sum_{i=1}^{n}|x(t_{i}) - x(t_{i-1})|^{p}$$ where $\Pi = \{a= t_{0}< t_{1}, \ldots, < t_{n} =b\}$ is a partition and $|\Pi| = \max_{i} |t_{i} - t_{i-1}|$ is its mesh size. It is well known that a Brownian motion is almost surely quadratic variation path in any subinterval of $[0,1]$, i.e. $V_{2}(B, [a, b]) \in (0,1)$ almost surely for arbitrary $0\le a < b \le 1$.

Q. Is there any simple and explicit construction of a single path, which has finite positive quadratic variation in any subinterval of $[0,1]$?

user79963
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    you have to be careful how you define quadratic variation to get BM to have that qv. With the ordinary definition, you get infinity. – Mark Joshi Jul 02 '15 at 08:40
  • @MarkJoshi May I have your more detailed explanation? I am not sure what you mean by ordinary definition? – user79963 Jul 02 '15 at 09:30
  • http://math.stackexchange.com/questions/92938/quadratic-variation-of-brownian-motion – Mark Joshi Jul 02 '15 at 10:32
  • @MarkJoshi Thanks for the link. In the above, we restrict the partition with its maximum mesh size converging to zero, i.e. $|\Pi| \to 0$. This avoids making the sum to be infinity. – user79963 Jul 05 '15 at 14:12

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