= Solution
Let
$$
A_t=\int_0^t\delta^2Y_u^{2(\delta-1)/\delta}du=\int_0^t\delta^2|B_u|^{2\delta-2}du.
$$
The clock $A$ is an <absolutely continuous function> and is strictly increasing: its derivative is positive away from the <Brownian zero set>, which has zero <Lebesgue measure>. It also tends to infinity. This is immediate for $\delta=1$; for $\delta>1$, recurrence and the <Strong Markov property> imply that the <Brownian occupation time> of, for example, $\{x:1<|x|<2\}$ is unbounded, while the integrand is bounded below there by $\delta^2$.
Thus $A$ is continuous, strictly increasing, and maps $[0,\infty)$ onto itself. Its <inverse function> $\tau_s=A^{-1}(s)$ is finite, continuous, and strictly increasing.
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