TITLE: Optional quadratic variation and predictable quadratic variation
QUESTION [7 upvotes]: What's the difference between optional quadratic variation (which sometimes is denoted by $
[M]$) and predictable quadratic variation (i.e $\ < M > $) of a stochastic process?

REPLY [12 votes]: Hi here are my two cents :
$<>$ is the predictable compensator of $[]$ process ( so the difference of both is a local martingale).
The difference is best illustrated by examining a Poisson process of intensity $\lambda$. 
Then the $[N]_t=\sum_{s\le t}(\Delta N_s)^2=\sum_{s\le t}(\Delta N_s)=N_t$ this process is not predictable as its jumps are inaccessible.
But $<N>_t=\lambda.t$ (comes from standard calculations) this process is predictable as it is deterministic.
And finally $[N]_t-<N>_t=N_t -\lambda.t$ is a (local) martingale.
You should take a look at George Lowther fantastic Blog "almost sure" 
Best regards