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Math Help - martingale proof

  1. #1
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    martingale proof

    I need to do the proof for the second condition but I'm not getting the right answer.

    \{S_n , n\geq 1\}} is a martingale if

    1) E[| S_n|]< \infty \forall n
    2) E[ S_{n+1}|S_1,...,S_n] = S_n \forall n

    S_n= X_1X_2...X_n

    Suppose E[| X_n|] < \infty \forall n and E[ X_n]=1

    Prove that \{S_n , n\geq 1\}} is a martingale



    I can prove the first condition but for the second one I keep getting E[ S_{n+1}|S_1,...,S_n] = S_n+1 which is wrong. Can anyone shine a light on this problem? Thanks

    (ps sorry for any bad use of LaTex, this is the first time I have used it!)
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  2. #2
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by willowtree View Post
    I need to do the proof for the second condition but I'm not getting the right answer.

    \{S_n , n\geq 1\}} is a martingale if

    1) E[| S_n|]< \infty \forall n
    2) E[ S_{n+1}|S_1,...,S_n] = S_n \forall n

    S_n= X_1X_2...X_n

    Suppose E[| X_n|] < \infty \forall n and E[ X_n]=1

    Prove that \{S_n , n\geq 1\}} is a martingale



    I can prove the first condition but for the second one I keep getting E[ S_{n+1}|S_1,...,S_n] = S_n+1 which is wrong. Can anyone shine a light on this problem? Thanks

    (ps sorry for any bad use of LaTex, this is the first time I have used it!)
    If this is a product and not a sum, then

    E[ S_{n+1}|S_1,...,S_n] = E( X_1X_2\cdots X_{n+1}|S_1,...,S_n)=X_1X_2\cdots X_nE(X_{n+1}|S_n)

    and I'm guessing you need E(X_1)=1

    YUP, I see it there.
    I found your E(X_1)=1

     =X_1X_2\cdots X_nE(X_{n+1}) =S_n (1)=S_n
    Last edited by matheagle; May 5th 2010 at 06:05 PM.
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  3. #3
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    ahh i seee, thank you, i was doing something a bit backwards with the stuff inside the brackets!
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