Assume you have a stochastic process $X_t$. Assume that the trajectories of X have finite variation on finite intervals.
Let $V(t,\omega)=\sup_\Pi\sum|X(t_{k+1},\omega)-X(t_k,\omega)|$, where $\Pi$ is the collection of partitions of $[0,t]$. Since X has finite variation trajectories, V is finite for every $(t,\omega)$. But is $(V_t)_{t\geq 0}$ adapted if $(X_t)_{t\geq 0}$ is adapted?
If I in some way could have written $V_t$ as a limit of rationals, where the rationals were less than t, it would have been fine. But the problem is that this may not allways be the case, and the the partition sequence may have to vary for each $\omega$?
Is it possible to show that $V$ is adapted. And if not, is it possible to show it if X is also continuous?
Also, do we even know that $V_t$ is a random variable, that is, is it measurable to the original $\sigma$-algebra, on the space where the process X is defined?
Fix some $t\in\mathbb{R}_{\geq 0}$ and define $Y_{n,k}=X_{tk/n}-X_{t(k-1)/n}$ for $n,k\in\mathbb{N}$ and $t\in\mathbb{R}_{\geq 0}$. Consider $$\lim_{n\rightarrow\infty}\sum_{k=1}^n|X_{tk/n}-X_{t(k-1)/n}|$$ For all $n\in\mathbb{N}$ and $k\in\{1,\ldots,n\}$, we have that $Y_{n,k}$ is measurable w.r.t. $\mathcal{F}_t$ since the sum of two random variables are measurable and so is $|Y_{n,k}|$ as a composition of two measurable functions and finally $\lim_{n\rightarrow\infty}\sum_{k=1}^n|Y_{n,k}|$ as the limit of measurable functions. It remains to prove that $V(t,\omega)=\lim_{n\rightarrow\infty}\sum_{k=1}^n|X_{tk/n}(\omega)-X_{t(k-1)/n}(\omega)|$ holds which follows from this question.