= Solution
As printed, the requested conclusion is false for $\delta\ne1$. On an interval on which $X$ stays positive, the <time change of a continuous process> satisfies $d\tau_s=ds/(\delta^2X_s^{2(\delta-1)/\delta})$. The time-changed martingale term has <quadratic variation> $s$, so the <Lévy characterization of Brownian motion> identifies it with a standard Brownian motion $\widetilde B$. Dividing the drift in part (a) by the derivative of the clock gives
$$
\frac{\frac{\delta(\delta-1)}2X_s^{(\delta-2)/\delta}}{\delta^2X_s^{2(\delta-1)/\delta}}=\frac{\delta-1}{2\delta X_s}.
$$
Consequently the construction actually satisfies
$$
dX_s=\frac{\delta-1}{2\delta X_s}ds+d\widetilde B_s,
$$
which is the <Bessel process> equation of dimension $2-1/\delta$. It equals the paper's claimed drift $(\delta-1)/(2X_s)$ only when $\delta=1$. The mismatch between the specified power, clock, and conclusion is therefore a typographical error in the question.
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