 Scale transform for an additive-noise diffusion

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 $g^{\prime \prime }=-2b{g}^{\prime }$. The Itô formula removes the drift of $g(X)$, giving $dY=h(Y)dW$, where $h=g^{\prime }\circ g^{-1}$ on the interval $g(\mathbb{R})$. If the drift is bounded, $h^{\prime }=-2b\circ g^{-1}$ is bounded, so $h$ is Lipschitz on that interval.