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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- Sep 23rd 2007, 02:08 AMBruceBronsonbasic 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. - Sep 23rd 2007, 06:04 AMCaptainBlack
Look at the definition of a Wiener Process then:

Group the terms together into non-overlapping time pairs:

$\displaystyle

W_0 + W_2 - W_3 + 2W_4 = W_0+W_2+W_3 + 2(-W_3+W_4)

$

......... $\displaystyle

= W_0+2W_2+(-W_2+W_3) + 2(-W_3+W_4)

$

......... $\displaystyle

= 3W_0+2(-W_0+W_2)+(-W_2+W_3) + 2(-W_3+W_4)

$

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 $\displaystyle 3W_0=0$ and

variance $\displaystyle 7$

RonL