# decomposition of r.v.

• Nov 5th 2011, 01:04 AM
Juju
decomposition of r.v.
Hallo everybody,

does anyone know, if there exists a decomposition of the random variable $X$ such that
$X=Z+Y$
where
$X$ is $\mathcal{F}-$measurable
$Z, Y$ are $\mathcal{G}-$measurable
$\mathcal{G}$ and $\mathcal{F}$ are two $\sigma-$algebras with $\mathcal{F} \subset \mathcal{G}$

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.

• Nov 5th 2011, 03:17 AM
Moo
Re: decomposition of r.v.
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.
• Nov 5th 2011, 12:41 PM
Juju
Re: decomposition of r.v.
$X,Y,Z$ aren't martingales ( $\mathcal{F}$ and $\mathcal{G}$ are no filtrations).
Sorry, I've forgotten to mention, that $Y,Z$ are not measurable with respect to $\mathcal{F}$.

Thank you.
• Nov 5th 2011, 04:48 PM
Moo
Re: decomposition of r.v.
Okay, here is a guess (I didn't entirely check if it works)

Consider $\mathcal H$ to be the set of all elements belonging to $\mathcal G$ and not belonging to $\mathcal F$. It can be proved that $\mathcal H$ is a sigma-algebra. Hence we can consider the conditional expectation of X with respect $\mathcal H$.
So if you take $Y=E[X|\mathcal H]$ and $Z=X-E[X|\mathcal H]$, it should work.