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Math Help - 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 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.

    Thanks in advance!
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  2. #2
    Moo
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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.

    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.
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  4. #4
    Moo
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    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.
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