 Solution

Since $H$ is continuous, strictly increasing and unbounded, $J_t=\inf \{s>0:H_s>t\}$ is its continuous inverse: $H_{J_t}=t$, $J_{H_s}=s$ and $J_0=0$. These inverse times are stopping times. Put $X_t=Y_{J_t}$, which is continuous and strictly positive and is adapted to the inverse-clock filtration ${\mathcal{G}}_t={\mathcal{F}}_{J_t}$.

Evaluate the identity from part (a) at $J_t$. The Lebesgue-Stieltjes change of variable $r=H_s$ givesThereforeThe process $\beta$ is Brownian motion in this same filtration by the construction in part (a). The drift integral is finite on each compact interval because the positive continuous process $X$ has a positive minimum there. This is the dimension-three case of the Exponential Brownian-to-Bessel time change.