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Math Help - basic weiner process

  1. #1
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    basic weiner process

    Can someone help me with this, i need to find the distribution of

    W_0 + W_2 - W_3 + 2W_4 where W is the standard weiner process

    Any help would be greatly appreciated thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by BruceBronson View Post
    Can someone help me with this, i need to find the distribution of

    W_0 + W_2 - W_3 + 2W_4 where W is the standard weiner process

    Any help would be greatly appreciated thanks.
    Look at the definition of a Wiener Process then:

    Group the terms together into non-overlapping time pairs:

    <br />
W_0 + W_2 - W_3 + 2W_4 = W_0+W_2+W_3 + 2(-W_3+W_4) <br />

    ......... <br />
= W_0+2W_2+(-W_2+W_3) + 2(-W_3+W_4) <br />

    ......... <br />
= 3W_0+2(-W_0+W_2)+(-W_2+W_3) + 2(-W_3+W_4) <br />

    Then the three bracketed terms are zero mean Guassian distributed and independent,
    with variances 4, 1 and 2. So the required distribution is Gaussian with mean 3W_0=0 and
    variance 7

    RonL
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