Given a sequence $\Pi=\{\pi_n\}$ of partitions of an interval $[0,T]$ with $|\pi_n|=\max\limits_{t^n_i,t^n_{i+1}\in \pi^n}|t^n_{i+1}-t^n_i|\to_{n\to +\infty} 0$ the quadratic variation of a path $x\colon [0,T]\to \mathbb{R}$ along $\Pi$ is defined by $$ [x]_{\Pi}(t)=\lim_{n\to +\infty}\sum_{\pi_n}|x(t_{i+1}\wedge t)-x(t_i\wedge t)|^2. $$
Let $\Pi'=\{\pi'_n\}$ be another set of partitions with $\pi'_n\subset \pi_n,$ $|\pi'_n|\to_{n\to+\infty} 0$.
My question: is it possible that while $[x]_{\Pi}$ exists and is finite $[x]_{\Pi'}$ does not exist, or if both $[x]_{\Pi},\, [x]_{\Pi'}$ exist do we necessarily have $[x]_{\Pi}=[x]_{\Pi'}$?
Note that this is not true without the condition $\pi'_n\subset \pi_n$. For example one can construct a set of partitions $\Pi'=\{\pi'_n\}$ with $|\pi'_n|\to_{n\to+\infty} 0$ so that the paths of Brownian motion a.s. have infinte quadratic variations along $\Pi'.$
I assume that $x$ is continuous ( or càdlàg “right continuous with left limits”).
In particular if we let $$ [x](t)=\lim_{n\to +\infty}\sum_{i=0}^{2^n-1}\left|x\left(\frac{i+1}{2^n}\wedge t\right)-x\left(\frac{i}{2^n}\wedge t\right)\right|^2. $$
Is it true that $$ \lim_{n\to +\infty}\sum_{k=1}^{2^{n-1}-1}\left|x\left(\frac{2k+1}{2^n}\wedge t\right)-x\left(\frac{2k-1}{2^n}\wedge t\right)\right|^2=[x](t)? $$