Lamperti transform (diffusion, $F^{\prime }(x)=1/\sigma (x)$)

On an interval where $\sigma >0$ is continuously differentiable, define $F(x)={\int }^x\sigma (y{)}^{-1}dy$. The Itô formula transforms $dX=b(X)dt+\sigma (X)dB$ intoThe transformed Itô diffusion has constant noise coefficient. For a power diffusion $\sigma (x)=x^{\alpha }$, $F(x)=x^{1-\alpha }/(1-\alpha )$ when $\alpha \ne 1$, and $F(x)=\log x$ when $\alpha =1$. This simplifies comparison with Brownian motion and exposes the boundary drift.