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The Girsanov's theorem is making me all confused. In my course literature they explain it by some simple discrete examples of coin-tossing etc. Saying that $Z$ is the ratio of $\frac{P^a(A)}{P(A)}$ when going from some probability measure $P^a$ to $P$. But then they do it in the case of Brownian motion, saying that this ratio would be $exp(-\theta B_n - \frac{1}{2} \theta^2 n)$. This choice doesn't make sense to me in the sense of being this ratio of probabilities, I mean I understand the proof that this choice of Z works, but how did they come up with this? Was it just that they saw that it worked and prooved it or is there any logic behind it?!

Thanks in advance!

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The most constructive is : Take $X_1,..,X_n$ i.i.d normal and calculate the ratio of the joint density when they have mean $\mu$ vs. mean $0$, and you will find it has the same form. So it is not a ratio probabilities but of densities. –  mike Dec 8 '12 at 14:30
    
Oh, I see. Ok that makes more sense. Thank you! –  Good guy Mike Dec 8 '12 at 14:34

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