= Scale transform for an additive-noise diffusion
{title2=$g'(x)=\exp(-2\int_0^x b(u)\,du)$}
For $dX=b(X)\,dt+dW$ with continuous drift, integrate the positive derivative displayed above. The resulting <scale function of a one-dimensional diffusion> solves $g''=-2bg'$. The <Itô formula> removes the drift of $g(X)$, giving $dY=h(Y)\,dW$, where $h=g'\circ g^{-1}$ on the interval $g(\mathbb R)$. If the drift is bounded, $h'=-2b\circ g^{-1}$ is bounded, so $h$ is Lipschitz on that interval.
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