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Math Help - conditional expectation

  1. #1
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    conditional expectation

    Let Y be a positive or integrable randoma variable on the probability space( \Omega  A P).
    If |Y| \leq C, show that |E(Y|  g )| \leq C, where  g is a sub \sigma algebra of  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
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    Quote Originally Posted by Moo View Post
    If Y is g-measurable, then it's A-measurable.
    Correct. But Y is A-measurable, not g-measurable...

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

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

    E[Y|g]=E[Y_+|g]-E[Y_-|g]\leq E[Y_+|g]\leq E[|Y||g]
    and
    -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
    |E[Y|g]|\leq E[|Y||g].

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