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Thread: conditional expectation

  1. #1
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    conditional expectation

    Let Y be a positive or integrable randoma variable on the probability space($\displaystyle \Omega$ $\displaystyle A $ $\displaystyle P$).
    If |Y| $\displaystyle \leq$ C, show that |E(Y|$\displaystyle g $)|$\displaystyle \leq$ C, where $\displaystyle g $ is a sub $\displaystyle \sigma$ algebra of $\displaystyle A $.
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  2. #2
    Moo
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    Hello,

    If Y is g-measurable, then it's A-measurable.
    And thus E(h(Y)|g)=h(Y), for any g-measurable function h...
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  3. #3
    MHF Contributor

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    Quote Originally Posted by Moo View Post
    If Y is g-measurable, then it's A-measurable.
    Correct. But $\displaystyle Y$ is A-measurable, not g-measurable...

    The inequality $\displaystyle |E[Y|g]|\leq E[|Y| |g]$ will enable to conclude by monotonicity of conditional expectation: $\displaystyle E[|Y||g]\leq E[C|g]=C$.

    This inequality can be seen as an instance of the conditional Jensen inequality ($\displaystyle x\mapsto|x|$ is convex), or more elementarily as follows: denoting $\displaystyle Y_+,Y_-$ the positive and negative parts of $\displaystyle Y$,

    $\displaystyle E[Y|g]=E[Y_+|g]-E[Y_-|g]\leq E[Y_+|g]\leq E[|Y||g]$
    and
    $\displaystyle -E[Y|g]=E[Y_-|g]-E[Y_+|g]\leq E[Y_-|g]\leq E[|Y||g]$
    (each of these inequalities holds by monotonicity of conditional expetation) hence
    $\displaystyle |E[Y|g]|\leq E[|Y||g]$.

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