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Thread: decomposition of r.v.

  1. #1
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    decomposition of r.v.

    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!
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  2. #2
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    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.
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  3. #3
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    Re: decomposition of r.v.

    $\displaystyle X,Y,Z$ aren't martingales ($\displaystyle \mathcal{F}$ and $\displaystyle \mathcal{G}$ are no filtrations).
    Sorry, I've forgotten to mention, that $\displaystyle Y,Z$ are not measurable with respect to $\displaystyle \mathcal{F}$.

    Thank you.
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  4. #4
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    Re: decomposition of r.v.

    Okay, here is a guess (I didn't entirely check if it works)

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