Hallo everybody,

does anyone know, if there exists a decomposition of the random variable $\displaystyle X$ such that

$\displaystyle X=Z+Y$

where

$\displaystyle X$ is $\displaystyle \mathcal{F}-$measurable

$\displaystyle Z, Y$ are $\displaystyle \mathcal{G}-$measurable

$\displaystyle \mathcal{G}$ and $\displaystyle \mathcal{F}$ are two $\displaystyle \sigma-$algebras with $\displaystyle \mathcal{F} \subset \mathcal{G}$

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.

Thanks in advance!