Let $\widehat B_t=rB_{t/r^2}$. By <Brownian scaling>, $\widehat B$ is standard Brownian motion. Substituting $Y_t=rX_{t/r^2}$ into the Bessel equation gives
$$
dY_t
=d\widehat B_t+\frac{d-1}{2Y_t}dt,
\qquad
Y_0=rx.
$$
Weak 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>.

Solution

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

{c}

If $X$ is a Bessel process of dimension $d$ started at $x$ and $r>0$, then
$$
(rX_{t/r^2})_{t\geq0}
$$
is a Bessel process of dimension $d$ started at $rx$. This follows from <Brownian scaling> in the stochastic differential equation and <uniqueness in law>.

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