= Lamperti transform
{disambiguate=diffusion}
{c}
{title2=$F'(x)=1/\sigma(x)$}
= Lamperti transform
{c}
{synonym}
On an interval where $\sigma>0$ is continuously differentiable, define $F(x)=\int^x\sigma(y)^{-1}dy$. The <Itô formula> transforms $dX=b(X)dt+\sigma(X)dB$ into
$$
dF(X_t)=dB_t+\left(\frac{b(X_t)}{\sigma(X_t)}-\frac12\sigma'(X_t)\right)dt.
$$
The transformed <Itô diffusion> has constant noise coefficient. For a <power diffusion> $\sigma(x)=x^\alpha$, $F(x)=x^{1-\alpha}/(1-\alpha)$ when $\alpha\ne1$, and $F(x)=\log x$ when $\alpha=1$. This simplifies comparison with <Brownian motion> and exposes the boundary drift.
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