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:
$\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