Scale transform for an additive-noise diffusion ($g^{\prime }(x)=\exp (-2{\int }_0^{x}b(u)du)$)

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 $g^{\prime \prime }=-2b{g}^{\prime }$. 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.

The zero extension of a scale diffusion coefficient at finite endpoints makes the transformed equation globally Lipschitz. Inverting its solution gives the original additive-noise equation. The pathwise bound $|X_t|\le |x|+{\sup }_{s\le T}|W_s|+\Vert b{\Vert }_{\infty }T$ prevents the inverse scale from reaching infinity at any finite time. This proves global strong existence and pathwise uniqueness even when the original drift is not Lipschitz.

The diffusion coefficient obtained from a scale transform for an additive-noise diffusion need only be defined on an open interval. If it is Lipschitz, its limit at a finite endpoint is zero: a positive limit would bound the derivative $1/h$ of the inverse scale and prevent the inverse from diverging. Extending it by zero beyond finite endpoints therefore preserves global Lipschitz continuity.