If you have a SDE say:

$dX = \mu dt + \sigma d W$,

where $\mu$ and $\sigma$ are not necessarily constant. Say, this process corresponds to a probability $P$. If you apply a Girsanov transformation and obtain a process $d\hat{X}$, under a new probability $\hat{P}$, is this new probability density always that of a Brownian motion density? I.e,. something like:

$\frac{1}{\sqrt{t} \sqrt{2 \pi}} exp \left[-\frac{1}{2} \left(\frac{x}{\sqrt{t}}\right)^2\right] dx$,

where this is the density of $d\hat{W}(t)$ at $x$?


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