Hello,
I don't quite understand your question... You're quoting decompositions of martingales, but is X,Y or Z a martingale ?
If not, your question is trivial, since a F-measurable rv will be G-measurable : consider Z=X and Y= 0.
Hallo everybody,
does anyone know, if there exists a decomposition of the random variable such that
where
is measurable
are measurable
and are two algebras with
I just know the decomposition of stochastic processes (continuous in time), such that a (local) martingale remains a (local) martingale in the enlarged filtration (initial enlargement of filtration). The Doob-Meyer Decomposition doesn't work here either.
Thanks in advance!
Hello,
I don't quite understand your question... You're quoting decompositions of martingales, but is X,Y or Z a martingale ?
If not, your question is trivial, since a F-measurable rv will be G-measurable : consider Z=X and Y= 0.
Okay, here is a guess (I didn't entirely check if it works)
Consider to be the set of all elements belonging to and not belonging to . It can be proved that is a sigma-algebra. Hence we can consider the conditional expectation of X with respect .
So if you take and , it should work.