# Explanation of the Girsanov's transformation

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?!

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