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Math Help - To prove martingale

  1. #1
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    To prove martingale

    I have problem in the following question

    Let \eta_1,...,\eta_n be a sequence of independent identically distributed random variables with E\eta_i=0.
    Show that the sequence \xi=(\xi_k) with

    \xi_k = \frac{exp \lambda.\left(\eta_1+...+\eta_k \right)}{\left(E\ \textrm{exp}\ \lambda \eta_1 \right)^{k}}
    is a martingale.

    I need to prove E(\xi_{k+1}|F_k)=\xi_k

    \xi_{k+1} = \frac{exp \lambda.\left(\eta_1+...+\eta_{k+1} \right)}{\left(E\ \textrm{exp}\ \lambda \eta_1 \right)^{k+1}}

    It can be decomposed as

    \xi_{k+1} = \frac{exp \lambda . \left(\eta_1+...+\eta_k \right)}{\left(E\ \textrm{exp}\lambda \eta_1 \right)^{k}}.\frac {\textrm{exp} \lambda\eta_{k+1}}{\left(E\ \textrm{exp}\ \lambda \eta_1 \right)}

    \xi_{k+1}=\xi_k.\frac {\textrm{exp} \lambda\eta_{k+1}}{\left(E\ \textrm{exp}\ \lambda \eta_1 \right)}

    E(\xi_{k+1}|F_k)=\xi_k.E(\frac {\textrm{exp} \lambda\eta_{k+1}}{\left(E\ \textrm{exp}\ \lambda \eta_1 \right)}|F_k)

    Anybody could help to bring
    E(\frac {\textrm{exp} \lambda\eta_{k+1}}{\left(E\ \textrm{exp}\ \lambda \eta_1 \right)}|F_k) =1
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  2. #2
    Moo
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    Hello,

    Since E[\exp(\lambda \eta_1)] is a constant, we can pull it out from the conditional expectation and we get \frac{1}{E[\exp(\lambda \eta_1)]}\cdot E\left[\exp(\lambda \eta_{k+1})\mid \mathcal F_k\right]

    Since (\eta_k) is a sequence of independent rv's, \eta_{k+1} and \mathcal F_k are independent, and so is \exp(\lambda \eta_{k+1}) (since it's a measurable function of \eta_{k+1}.

    Thus we have \frac{1}{E[\exp(\lambda \eta_1)]} \cdot E[\exp(\lambda \eta_{k+1})] and since the \eta_k have the same distribution, these two expectations are equal...
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