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Math Help - Random Processes (Martingale)

  1. #1
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    Random Processes (Martingale)

    Hello all, I have a bit of a problem at the moment solving this task. I done half of it already, just can solve it for the other half. Hope u can help! Thanks =)

    Let S_n = e^{\sum Y_i} where Y_i, i = 1, 2,...,n all are independent with N (\mu, \sigma) distribution, find a relation such that r can be expressed in terms of \mu, \sigma^2

    Here's my working out so far:

    Def of a martingale: E(X_{n+1} | X_1,...,X_n) = X_n

    Hence: E (S_{n+1} e^{-r(n+1)} | S_1,,,,,S_n) = S_ne^{-rn}

    Now because S_n = e^{\sum Y_i} where Y_i, i = 1, 2,...,n

    That becomes:  E(e^{Y_1 + ....+ Y_{n+1}} e^{-r(n+1)} |S_n)

    From that point im not too sure what to do. If anyone can help me or tell me what to do would be much appreciated. Thanks.
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  2. #2
    Moo
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    Hello,

    What is r ?
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    "r" is just a constant variable.
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    Yes, but you're not clear at all !!!!!

    I'll assume you want to find r such that S_ne^{-rn} is a martingale. If that's not the case, then read again what you wrote...

    E(e^{Y_1 + ....+ Y_{n+1}} e^{-r(n+1)} |S_n)
    Okay with this.

    Now this can also be written : e^{-rn} e^{-r} \cdot E[e^{Y_1+\dots+Y_n}e^{Y_{n+1}}\mid S_n]

    Since the Yn are independent, we'll have a product of expectations :

    e^{-rn} \cdot E[e^{Y_1+\dots+Y_n}\mid S_n] \cdot e^{-r} \cdot E[e^{Y_{n+1}}\mid S_n]

    Since Y1,...,Yn are Sn-measurable, E[e^{Y_1+\dots+Y_n}\mid S_n]=e^{Y_1+\dots+Y_n}

    Since Y_{n+1} is independent with S_n, E[e^{Y_{n+1}}\mid S_n]=E[e^{Y_{n+1}}]


    So finally, you want r such that e^{-rn} \cdot e^{Y_1+\dots+Y_n}\cdot e^{-r} \cdot E[e^{Y_{n+1}}]=S_n e^{-rn}

    which means to find r such that e^{-r}\cdot E[e^{Y_{n+1}}]=1

    Have a look at the MGF of a normal distribution when t=1 and you're done...


    (to simplify a bit all these things, you could've written S_{n+1} in terms of S_n, but it doesn't really matter, it's just more beautiful)
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    i think that should be it! Thanks alot!
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