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The definition of quadratic variation of a stochastic process $Y$ is that $[Y]_t$ for which for each $\epsilon>0$ there exists a $\delta>0$ such that $$P\left(\left|\sum_{i=0}^{m(\pi)-1}(Y_{t_{i+1}}-Y_{t_i})^2-[Y]_t\right|\geq\epsilon\right)\leq\epsilon$$ for all partitions $\pi$ with $\operatorname{mesh}(\pi)\leq\delta$.

The following 'definition' seems much more natural to me, is it equivalent?

For all $t\geq 0$ and almost all $\omega\in\Omega$ $$[Y]_t(\omega)=\sup_{\text{partitions }\pi}\sum_{i=0}^{m(\pi)-1}(Y_{t_{i+1}}(\omega)-Y_{t_i}(\omega))^2$$ I wonder this, because I cannot find a definition of quadratic variation outside the context of stochastic processes, while the above can be defined as the quadratic variation of the function $Y(\omega)$.

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    $\begingroup$ No, it's not equivalent... $\endgroup$ – saz May 30 '17 at 14:30

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