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?
Optional quadratic variation and predictable quadratic variation
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stochastic-processes
quadratic-variation
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1See also this question: http://math.stackexchange.com/q/902886/ – 2015-03-23
1 Answers
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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 $
And finally $[N]_t-
You should take a look at George Lowther fantastic Blog "almost sure"
Best regards