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Let $(X_t)$ be a semimartingale. I read that its quadratic variation can be written as

$$ [X]_t= X_0+ [X]_t^c + \sum_{0\leq s \leq t} (\Delta X(s))^2$$

What does $[X]_t^c$ mean ? Is it the quadratic variation of $(X_t)$ if it was continuous, i.e the quadratic variation of $X_t - \sum_{0\leq s \leq t} \Delta X(s)$? I don't believe this to be true since we would need to assume that $\sum_{0\leq s \leq t} |\Delta X(s)|$ is finite, which is not the case for all semimartingales.

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This is how Protter defines $\left[X\right]^c_t$ (Section II.6):

Definition: For a semimartingale $X$ the process $\left[X\right]^c_t$ denotes path-by-path continuous part of $\left[X\right]$.


Note that analogously, one can define $[X, Y]^c$ as the path-by-path continuous part of $[X, Y]$.

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