Converse to Girsanov's theorem? Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to adaptedness constraints on the drift, integrability, etc...).
I'm wondering if there's a 'converse' to Girsanov's theorem. If we have some process $X$ whose law $\mathbb{P}$ is equivalent to Wiener measure $\mathbb{W}$, can we always express $X$ as $X_t = \alpha_t + W_t$, where $\alpha$ is adapted to the filtration of $W$? Alternatively, can the Radon-Nikodym derivative always be expressed in the form
$$\frac{d\mathbb{P}}{d\mathbb{W}} = \exp\left( \int \alpha_u dW_u - \frac{1}{2}\int \alpha^2_u du  \right)$$.
 A: http://books.google.com.sg/books?id=YoMr5Mbo6coC&pg=PA164&lpg=PA164&dq=bjork+girsanov+theorem&source=bl&ots=imju5rQt6e&sig=Tp9bgNO6hDNZEXuEy6tmBs5nwZY&hl=en&sa=X&ei=HYkUUdaELoqKrgesioHYDA&redir_esc=y#v=onepage&q=bjork%20girsanov%20theorem&f=false
A: The answer is yes:
https://fabricebaudoin.wordpress.com/2012/10/02/lecture-25-girsanov-theore/
Here is a good link. I think it is a better link than the currently most upvoted answer.
A: Although I can't see the page linked by A.Kvashchuk, I presume it mostly answers your question along the lines: construct a positive martingale with the Radon-Nikodym derivative as terminal condition, represent it first as a stochastic integral and then (since it is positive) as the solution of a linear s.d.e. to yield a Doleans exponential.
In some cases, such as $X = -W$, this will yield $\alpha = 0$ , at odds with your equation $X = \alpha + W$.
It may help to clarify the terms in your question:


*

*defining $\alpha$ as $X-W$ does not get us very far, and needlessly requires to define the two processes on the same probability space.

*the Doleans exponential you show would correspond to $X = W - \int \alpha$ .

