For $Y=X^\alpha$, the <Itô formula> and the Bessel equation give
$$
dY_t=\alpha X_t^{\alpha-1}dB_t
+\frac{\alpha(d+\alpha-2)}2X_t^{\alpha-2}dt.
$$
Use the clock
$$
C_t=\alpha^2\int_0^tX_s^{2\alpha-2}ds
$$
and its inverse. The <Dambis-Dubins-Schwarz theorem> turns the first term into Brownian motion, while division of the drift by the clock rate gives
$$
d\widetilde Y_u=dW_u+
\frac{d+\alpha-2}{2\alpha}\frac{du}{\widetilde Y_u}.
$$
Hence $\widetilde Y$ is a Bessel process of dimension
$$
d'=1+\frac{d+\alpha-2}{\alpha}
=2+\frac{d-2}{\alpha}.
$$
This is the <Power time change of a Bessel process>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 4 / e / Solution

{c}

If $X$ is a Bessel process of dimension $d$ and $\alpha>0$, then $X^\alpha$, after the clock $\alpha^2\int X_s^{2\alpha-2}ds$, is a Bessel process of dimension
$$
d'=2+\frac{d-2}{\alpha}.
$$


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