Resources on an intuitive understanding of the Girsanov Transformations My current project involves the use of Girsanov transformation. Can anyone suggest me some resources for an intuitive understanding of the same. The pages I have been referring to (Wikipedia), deals with it very mathematically.
 A: The question does not exactly seem motivated by physics and probably without further clarification belongs to the math SE. However, let's below take a hand-wavy approach to the subject to build a bit of intuition, anyway, as a form of a long comment more so than a good answer to the question. Before doing that, though, I'll just comment briefly and say that I am not aware of any areas of physics that are not very mathematical where Girsanov really is of use (and if you are interested in mathematical physics, you'll probably want to read the mathematicians' more rigorous writeups, for example the introductory book by Øksendal, Stochastic Differential Equations, contains Girsanov's theorem).
Now, then, given a random variable $X$, what's the probability of getting a value $x$? The answer depends on the coordinate system.
Let's say we know the distribution $p^{\mathbb{P}}_X$ of the random variable $X$ under some coordinate system (or, measure) $\mathbb{P}$, and so its expected value is 
$$E^{\mathbb{P}}(X) = \int x\, p^{\mathbb{P}}_X(x) \,\mathrm{d}x.$$
We could equally well have changed to some other measure $\mathbb{Q}$: $$E^{\mathbb{P}}(X) = \int x\, p^{\mathbb{P}}_X(x) \,\mathrm{d}x = \int x\, \frac{p^{\mathbb{P}}_X(x)}{p^{\mathbb{Q}}_X(x)} p^{\mathbb{Q}}_X(x) \,\mathrm{d}x = E^{\mathbb{Q}}\left(X \frac{p^{\mathbb{P}}_X}{p^{\mathbb{Q}}_X}\right) = E^{\mathbb{Q}}\left(X \frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\right),$$
where we've defined the quantity (called the Radon-Nikodym derivative) $\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}$ as the ratio of the probability density functions. Of course we can't divide by zero, so this quantity only exists when the measures are equivalent, i.e. when the measures agree on what is possible, restricting the kind of coordinate transforms that can be done and have all this still make sense.
Before getting to Girsanov, there's one more important concept we need to introduce: martingales. Suppose we have a stochastic process $X_t$. It is a $\mathbb{P}$-martingale if for all $s < t$, 
$$E^\mathbb{P}_s(X_t) = X_s$$
where the subscript $s$ denotes an expectation conditioned on all knowledge up to $s$. Note then the Radon-Nikodym process $\varsigma_t = E_t^{\mathbb{Q}}\left(\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\right)$ is a martingale (by the tower law/iterated expectations):
$$E_s^\mathbb{Q}(\varsigma_t) = E_s^{\mathbb{Q}}\left(E_t^{\mathbb{Q}}\left(\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\right)\right) = E_s^{\mathbb{Q}}\left(\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\right) = \varsigma_s.$$
Now we take a $\mathbb{P}$-Brownian motion, and without trying to define it in a more rigorous sense, it is basically a sum (or, Ito integral) of i.i.d. Gaussian increments:
$W^{\mathbb{P}}_t = \int_0^t \mathrm{d}W^\mathbb{P}_s$, and this is normally distributed as $\mathcal{N}(0, t)$ under $\mathbb{P}$. Note that we can write any Ito process $X_t$ that is a $\mathbb{P}$-martingale as $\mathrm{d}X_t = \sigma(t, X_t, \omega)\, \mathrm{d}W_t^{\mathbb{P}}$: there can be no $\mathrm{d}t$ drift term.
Finally, the question Girsanov asks and answers, then, is "how is this distributed under an equivalent measure $\mathbb{Q}$?". 
We've already established that $\varsigma_t$ is a $\mathbb{Q}$-martingale, and so we must be able to write this as $\mathrm{d}\varsigma_t = \sigma(t, \varsigma_t, \omega) \,\mathrm{d}W_t^{\mathbb{Q}}$, but because $\varsigma_t$ as a ratio of two PDFs is always positive, we might as well write this in the conventional form: $\frac{\mathrm{d}\varsigma_t}{\varsigma_t} = -\gamma_t\, \mathrm{d}W_t^{\mathbb{Q}}$. Now with $\mathcal{F}$ the Fourier transform (and $k$ the frequency domain variable in the sloppy notation below):
$$p^{\mathbb{P}}_{W_t^{\mathbb{Q}}}(w) = \mathcal{F}^{-1}(\mathcal{F}(p^{\mathbb{P}}_{W_t^{\mathbb{Q}}}))(w) = \mathcal{F}^{-1}(\overline{E^{\mathbb{P}}(e^{ikW_t^{\mathbb{Q}}}}))(w) = \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{1}{2t}\left(w + \int_0^t\gamma_s\mathrm{d} s\right)^2\right),$$
which follows after tedious, though, straightforward computation. We see that the above is a Gaussian, and what happens in a change of measure, therefore, is that the variance is left unchanged, but we end up with a drift term, i.e. $\mathcal{N}(-\int_0^t\gamma_s\mathrm{d} s, t)$ under $\mathbb{Q}$.
I've simplified quite a few things and skipped over some aspects of definitions etc. (and I've only shown that the marginal is Gaussian, but marginals are not the be all end all to processes, after all for example $t\,\mathrm{d}W$ and $W\,\mathrm{d}t$ have the same marginals, but only one of them is a martingale), so I do recommend you consult e.g. the book by Øksendal for a more complete picture.
