= Power time change of a Bessel process
{c}
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> gives
$$
d(X_t^\alpha)=\alpha X_t^{\alpha-1}dB_t+\frac{\alpha(\delta+\alpha-2)}2X_t^{\alpha-2}dt.
$$
Use the strictly increasing clock $A_t=\alpha^2\int_0^tX_s^{2\alpha-2}ds$ 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 $\delta'=2+(\delta-2)/\alpha$ on this changed stochastic interval. If $\delta'$ 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.
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