Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 3 / c / Solution

Let ${\hat{B}}_t=r{B}_{t/r^2}$. By Brownian scaling, $\hat{B}$ is standard Brownian motion. Substituting $Y_t=r{X}_{t/r^2}$ into the Bessel equation givesWeak uniqueness for the Bessel equation shows that $Y$ is a Bessel process of dimension $d$ started at $rx$. This is the Scaling invariance of a Bessel process.