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Let $W(\omega,x):\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a Brownian motion where $\Omega$ is the sample space. Recall that the quadratic variation of $W$ over the interval $[a,b]$ equals $b-a$ almost surely, that is to say:

$$\langle W \rangle = \lim_{n\rightarrow\infty}\sum_{i=1}^{n} \left| W\left(\omega,a+\frac{i}{n}(b-a)\right)-W \left(\omega,a+\frac{i-1}{n}(b-a)\right) \right|^2 = b-a$$

almost surely.

Now consider some continuous function (this time not a random variable) $f:\mathbb{R}\rightarrow\mathbb{R}$ and consider the more general limit:

$$\langle f \rangle = \lim_{n\rightarrow\infty}\sum_{i=1}^{n} \left| f\left(a+\frac{i}{n}(b-a)\right)-f \left(a+\frac{i-1}{n}(b-a)\right) \right|^p $$

I would like to know if there exists a continuous function $f$ for which $\langle f \rangle$ is finite but non zero for a value of $p$ greater than 2. For example, does there exist a function $f$ of finite but non zero 'cubic variation' ($p=3$).

More generally, for which values of $p$ do such functions $f$ exist.

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    $\begingroup$ It's worth noting that $p$ variation is sometimes (usually?) defined using arbitrary partitions of the interval in question (as in Riemannian integration). While the limit you wrote for BM is true for that particular partition or any fixed sequence of partitions (at least, you get convergence in probability), BM almost surely has infinite quadratic variation --- if quadratic variation is defined using arbitrary partitions. $\endgroup$
    – user711689
    Jul 17, 2021 at 2:59

1 Answer 1

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The paths of fractional Brownian motion with Hurst exponent $H$ admit (almost surely) non-vanishing $\frac{1}{H}$-variation. Since $H \in (0,1)$, then you can find paths with your desired property for $p = \frac{1}{H} > 1$.

But, of course, that requires a probability model on path space, which is not what you want. We can still provide explicit paths that admit non-vanishing $p$-variation for any $p>1$. For simplicity, I'll change the partition on which you sample points on your path to the dyadic partition of $[0,1]$. We define the Takagi-Landsberg functions with Hurst parameter $H\in(0,1)$ by $$\mathcal{X}^H= \left\lbrace f \in C[0,1] \, : \, f = \sum_{m=0}^\infty 2^{m\left( \frac{1}{2} - H \right)} \sum_{k=0}^{2^m - 1} \theta_{m,k} e_{m,k} ,\, \theta_{m,k}\in\{-1,1\} \right\rbrace$$

Where $e_{m,k}$ are the Faber-Schauder basis functions for $C[0,1]$. Then, by Theorem 2.1 of [1], we find that if $f \in \mathcal{X}^H$ then the sequence of functions

$$\langle f\rangle_{t}^{(n)} = \sum_{\substack{k=0 \\ k2^{-n} \leq t }}^{2^n-1}|f((k+1)2^{-n}) - f(k2^{-n})|^{1/H}$$

converges uniformly to the linear function $\langle f\rangle_{t} = a_{H} t$, where $a_H$ is a constant depending only on $H$.

[1] Mishura, Yuliya, and Alexander Schied. "On (signed) Takagi–Landsberg functions: pth variation, maximum, and modulus of continuity." Journal of Mathematical Analysis and Applications 473.1 (2019): 258-272.

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