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Let $\Bbb D(\Bbb R_{\ge0},\Bbb R^d)$ be the space of the Càdlàg functions.

If $p\ge1$ and $x\in\Bbb D(\Bbb R_{\ge0},\Bbb R^d)$, we define the norm $$ \overline V_p(x)_T:=|x_0|+\left(\sup_{\pi}\sum_{j=1}^n|x_{t_j}-x_{t_{j-1}}|^p\right)^{1/p} $$

where $\pi$ varies through all the partions of the interval $[0,T]$ and $|\cdot|$ is the usual euclidean norm.

My problem is the following: is $$ \overline V_p(x)_T\le\sup_{s\in[0,T]}|x_s|\;\;\;? $$

I know that for example a.s. the Brownian motion has unbounded total variation but a.s. it is continous; so fixing a suitable $\omega\in\Omega$ we would have a deterministic function which seems to be a counterexample when $p=1$.

But: what can we say for $p\ge1$?

I can't neither prove nor disprove this statement.

Joe
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1 Answers1

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The function $f(x)=x\sin\frac1x$ with $f(0)=0$ is the typical example of a bounded continuous function for which the variation is infinite. See this link bounded not BV By modifying this function, with say $f(x)=x^{1/p}\sin\frac1x$, I think you get a counterexample for $p\ge 1$.

Gio67
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