Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 30 / 2 / c / Solution

Let $B=\Vert b{\Vert }_{\infty }$ and define the scale function of a one-dimensional diffusionSince $b$ is continuous, $g\in C^2$, andThus $g$ is strictly increasing. The Itô formula for the additive-noise equation givesThe integrand is locally square-integrable: a continuous path $X$ has compact range on every finite interval, where $g^{\prime }$ is bounded. HenceThis is the scale transform for an additive-noise diffusion. Its range is the open interval $I=g(\mathbb{R})$, which need not be all of $\mathbb{R}$.