 Power time change of a Bessel process

Let $X$ be a Bessel process of dimension $\delta$, started at $x>0$, and let $\alpha >0$. Work before the first hit of zero. The Itô formula givesUse the strictly increasing clock $A_t={\alpha }^2{\int }_0^t{X}_s^{2\alpha -2}ds$ and its inverse $\tau$. Then $Y_u=X_{{\tau }_u}^{\alpha }$ has unit noise coefficient and drift coefficient $(\delta +\alpha -2)/(2\alpha Y_u)$. It is therefore a Bessel process of dimension ${\delta }^{\prime }=2+(\delta -2)/\alpha$ on this changed stochastic interval. If ${\delta }^{\prime }$ is negative, the assertion means the positive diffusion killed at zero. No continuation through zero follows from this computation, where the clock or the power derivative can become singular.