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Work before the first zero of $Z$, with $Z_0=z>0$. For the Bessel process equation, the Itô formula applied to $x^r$ givesIts quadratic variation isThis clock is strictly increasing on the positive-state interval. For $s$ before its terminal clock value, let $Y_s=W_{{\sigma }_s}$ and let ${\beta }_s=({\int }_0^{\cdot }r{Z}_u^{r-1}d{B}_u{)}_{{\sigma }_s}$. The Dambis-Dubins-Schwarz theorem makes $\beta$ a Brownian motion there. Since $d{\sigma }_s=ds/(r^2{Z}_{{\sigma }_s}^{2r-2})$, the drift changes toConsequently the Power time change of a Bessel process givesThe initial value is $z^r$, and this equation identifies a Bessel process of dimension ${\delta }^{\prime }$ until its first zero. Localize both equations on compact subintervals of $(0,\infty )$ to justify the inverse and all stochastic integrals for arbitrary $r>0$.

For completeness, the inverse-clock lifetime really is the positive Bessel process lifetime: the map $z\mapsto z^r$ is a homeomorphism of $(0,\infty )$, and exits from nested compact intervals correspond under the clock. There is no finite explosion at infinity for a positive Bessel process, since the Itô formula for $Y^2$ gives $d(Y^2)=2Yd\beta +{\delta }^{\prime }ds$; stopping at zero or level $R$ yields a probability of hitting $R$ before a fixed $T$ bounded by $(Y_0^2+\max ({\delta }^{\prime },0)T)/R^2$. If the transformed lifetime were finite while $Y$ stayed in a compact positive interval, the inverse clock and the original solution could both be continued. Thus a finite lifetime is approach to zero. For $\delta <2$, also ${\delta }^{\prime }<2$ and the positive-state Bessel process hits zero; for $\delta \ge 2$ the transformed lifetime is infinite. No continuation through zero is needed.

If the initial value is already zero, the interval before its first zero is empty; the positive-state argument concerns starts $z>0$.

For $Y=X^{\alpha }$, the Itô formula and the Bessel equation giveUse the clockand its inverse. The Dambis-Dubins-Schwarz theorem turns the first term into Brownian motion, while division of the drift by the clock rate givesHence $\tilde{Y}$ is a Bessel process of dimensionThis is the Power time change of a Bessel process.