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Math Help - Martingales - Explaination.

  1. #1
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    Martingales - Explaination.

    Hi all!

    I've got a problem with a question at the moment and i was wondering if you could clarify it? Thanks =)

    Let X_{n} = X_{0} + \sum^n_{i=1} Y_{i} be a random walk.

    Suppose that t^* > 0 such that m(t^*) = 1 where m(t)=e^{tY_{i}} is the moment generating function of Y_{i}.

    Show e^{t^*X_{n}}

    The property of martingale:

    E|X_{n}| < \infty

    E(X_{n+1}|X_{1},X_{1}...X_{n})

    so show that : E(X_{n+1}|X_{1},X_{2}...X_{n}) = e^{t^*X_{n}}

    Now E(e^{t^*(X_{0}+Y_{1}+...+Y_{n}} |X_{1}...X_{n})

    From that it becomes

    E(e^{t^*X_{0}+t^*Y_{1}...+t^*Y_{i}}|X_{1}...X_{n})

    Now the one thing i dont get is that the equation then becomes:

    E(e^{t^*X_{n}}e^{Y_{i}}|X_{1}...X_{n})

    = e^{t^*X_{n}}E(e^{Y_{i}}|X_{1}...X_{n})

    why from suddenly e^{t^*X_{0}} becomes e^{t^*X_{n}}?

    Any explainations would be great. Thank you.

    =)
    Last edited by Redeemer_Pie; April 24th 2010 at 01:22 AM.
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  2. #2
    Moo
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    Hello,

    Because X_n=X_0+Y_1+\dots+Y_n ?

    Now E(e^{t^*(X_{0}+Y_{1}+...+Y_{n{\color{red}+1}}} |X_{1}...X_{n})

    From that it becomes

    E(e^{t^*X_{0}+t^*Y_{1}...+t^*Y_{\color{red}n+1}}|X  _{1}...X_{n})

    Now the one thing i dont get is that the equation then becomes:

    E(e^{t^*X_{n}}e^{{\color{red}t^*}Y_{\color{red}n+1  }}|X_{1}...X_{n})

    = e^{t^*X_{n}}E(e^{{\color{red}t^*}Y_{\color{red}n+1  }}|X_{1}...X_{n})
    I think there are typos (in red)
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  3. #3
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    Oh k. Thanks. lol that was a bit embarassing but i forgot to add:

    I thought via martingale property that it shouldnt we be solving for:

     E(e^{t^*X_{n+1}}|X_{1}...X_{n})

    rather than:

     E(e^{t^*X_{n}}|X_{1}...X_{n}) ?
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  4. #4
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    Quote Originally Posted by Moo View Post
    Hello,

    Because X_n=X_0+Y_1+\dots+Y_n ?


    I think there are typos (in red)

    yep! sorry they meant to say Y_{i}

    Sorry my first time using LaTeX
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  5. #5
    Moo
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    I don't understand... Do you have any further question ? oO
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  6. #6
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    nope that'll be all thanks
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