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In Diffusions, Markov Processes and Martingales Volume 2 by Rogers and Williams they state the following theorem due to Meyer:

$\mathbf{Theorem }$ Le $M\in\mathcal{M}^2_0$. Then there exists a unique increasing process $[M]$ null at $0$ such that

  1. $M^2-[M]$ is a uniformly integrable martingale
  2. $\Delta [M]=(\Delta M)^2$ on $(0,\infty)$

By definition $\mathcal{M}^2_0$ is the space of martingale null at $0$, which are bounded in $L^2$. I'm confused about the following: Clearly $[M]$ is adapted, by $1.$, but I also read (in a comment in my lecture notes) that it is even predictable. Why do Rogers and Williams not mention this? First I was also confused that they do not mention the adaptness explicitly. Usually they write something like this: "...exists a unique adapted increasing process...". I just want to be sure, that I do not miss something important here. Thanks for clarification.

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The result you cite regards the existence "ordinary" quadratic variation process, $[M]$, of a square-integrable martingale. It is adapted, but not in general predictable. To "compensate" for its lack of predictability and still ensure uniqueness, the requirement on the jumps is made. There is another process, $\langle M\rangle$, the predictable quadratic variation, which is characterized by having $M^2 - \langle M\rangle$ be a local martingale and having $\langle M\rangle$ be predictable (but no requirement on the jumps, in contrast to the theorem you cite). The process $\langle M\rangle$ is given as the dual predictable projection of $[M]$.

See for example "Semimartingale theory and stochastic calculus" by He, Wang and Yan for more on all processes mentioned above.

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