# Ornstein-uhlenbeck generator

• Aug 5th 2010, 07:27 AM
Sjaak
Ornstein-uhlenbeck generator
I woud like to show that the generator of the OU process is given by
$Af(x)=\frac{1}{2} \sigma^2f''(x)-\alpha x f'(x)$

i know generator of brownian motion Af=1/2f''(x) and it should be something like this:

first proof that
$P_{t}^{X}f(x)=P^{W}_{g(t)}f(e^{-\alpha t}x)$ with $P_{t}^{X}, P_{t}^{W}$ transitions functions of OU process and BM, $g(t)=\sigma^2(1-e^{-2\alpha t})/2\alpha$
and then (i guess) something like
$\frac{d}{dt}P_{t}f=AP_{t}f$

any help/hints would be appreciated! :)
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edit:
OU process is defined as $X_t=e^{-\alpha t}(X_0+W_{\frac{\sigma ^2}{2\alpha}(e^{2\alpha t}-1)$