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Thread: 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 $\displaystyle X_{n}$ = $\displaystyle X_{0}$ + $\displaystyle \sum^n_{i=1}$$\displaystyle Y_{i}$ be a random walk.

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

    Show $\displaystyle e^{t^*X_{n}}$

    The property of martingale:

    $\displaystyle E|X_{n}| < \infty$

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

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

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

    From that it becomes

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

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

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

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

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

    Any explainations would be great. Thank you.

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

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

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

    From that it becomes

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

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

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

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

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

    rather than:

    $\displaystyle 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 $\displaystyle X_n=X_0+Y_1+\dots+Y_n$ ?


    I think there are typos (in red)

    yep! sorry they meant to say $\displaystyle Y_{i}$

    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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