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Math Help - Expectation with Brownian motion

  1. #1
    Junior Member
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    Expectation with Brownian motion

    I need to find \mathbb{E}\left[e^{W(1)+W(2)}\right] , where W is a standard Brownian motion.

    I tried the following.

    W(1)+W(2)=(W(2)-W(1))+2W(1)

    Brownian motion has independent increments, and I should be able to write the above sum as two independent standard normal variables, i.e.

    (W(2)-W(1))+2W(1)=G_{1}+2G_{2} , where  G_1 and G_2 are independet standard normal variables.

    Thus...

    \mathbb{E}\left[e^{W(1)+W(2)}\right]= \mathbb{E}\left[e^{G_{1}+2G_2}\right]=\mathbb{E}\left[e^{G_1}\right] \cdot \mathbb{E}\left[e^{2G_2}\right] = e^{1/2} \cdot e^{2} = e^{\frac{5}{2}}

    In the last line I used the fact that independent random variables have factoring moment-generating function and also the fact that \mathbb{E}\left[e^{uG}\right] = e^{\frac{1}{2}u^2} .

    Is the above reasoning correct, if not, what is wrong? Thanks!
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  2. #2
    MHF Contributor
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    your reasoning up to the last line is fine.

    I have not seen the result you use on the last line so i can comment on that.
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