= Finite-horizon drift replacement by a change of measure
For $dX=\mu\,dt+\sigma\,dB$ on $[0,T]$, take $\theta=(\nu-\mu)/\sigma$ and use density $\mathcal E(\int\theta\,dB)_T$. If $\mu,\nu$ are bounded and $\sigma$ is continuous and bounded below by a positive constant, the exponential has bounded bracket and defines an equivalent measure. Under it, $\widetilde B=B-\int\theta\,dt$ is Brownian and $dX=\nu\,dt+\sigma\,d\widetilde B$. Set $\theta=0$ after $T$ to define the density on the entire original sigma-algebra and the new <Brownian motion> on the whole time axis.
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