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I am using Revous and Yor's book to learn about qudratic variations. For an arbitrary process this is defined via the following pointwise (in t) limit in probability

$T^{\Delta _{n}}_{t} =\sum_{i=0}^{k-1}(X_{t_{i+1}}-X_{t_{i}})^2+(X_{t}-X_{t_{k}})^2$

But when when we define a similar concept for a local martingale we instead have that the random variables

$\sup_{s\le t}\mid T^{\Delta _{n}}_{s}(M) - <M,M>_{s}\mid$ converging to zero in probablity.

The latter looks stronger since it have to be uniform in the sequence of finite partitions. However is still looks like the same sum converges, just in a stronger sense.

Yor do not however, call this the quadratic variation but he defines as the "increasing process of $M$". And furthermore for distinct local martingales he calls it the "bracket".

Is this common pratice not to call this stronger limit the qudratic variation but something else? And if yes is there a particular reason for this?

My own guess is that this might have something to do with the fact that this is only considered for martingales and not processes in general, making it a subclass of the more general quadratic varitaion. They do however use citation marks in connection to the following statement

"Brownian motion is not a bounded martingale, nevertheless we have seen it has "qudratic variation" $t$".

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The usual notation for quadratic variation is $[\cdot]$.

You may also define, for a local martingale $M$, a process that is sometimes called the conditional quadratic variation: $\langle M,M\rangle$ is the nondecreasing process such that $M^2-\langle M,M\rangle$ is a local martingale. Revuz and Yor are referring to this process as the "increasing process associated with $M$." And they're showing, in the same theorem (i.e., Theorem IV.1.8), that the "increasing process associated with $M$" is the same as the quadratic variation process.

The quadratic variation process $[\cdot]$ and the bracket process $\langle\cdot\rangle$ are the same for continuous local martingales, so you don't have to worry about the distinction. But they may be different for discontinuous local martingales. See Angle bracket and sharp bracket for discontinuous processes

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