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

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.