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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.

The Itô formula for $f(t,x)=e^{x+at}$ gives the stochastic differential equationWrite $N_t={\int }_0^t{Z}_{u}d{B}_u$ for its continuous local martingale part. Its clock is the quadratic variationBecause $Z_t>0$, $A$ is strictly increasing. For $s<A_{\infty }$, its inverse ${\tau }_s$ satisfies $A_{{\tau }_s}=s$ and $d{\tau }_s/ds=Z_{{\tau }_s}^{-2}$. The Dambis-Dubins-Schwarz theorem, or the Lévy characterization of Brownian motion after the optional time-change theorem, makes ${\beta }_s=N_{{\tau }_s}$ a Brownian motion for the time-changed filtration. With $Y_s=Z_{{\tau }_s}$,This is the Bessel process equation with dimension $\boxed{2+2a}$, on the positive stochastic interval. This establishes the Exponential Brownian-to-Bessel time change.

The lifetime convention is important: the inverse is defined only for $s<A_{\infty }$. If $a<0$, the strong law for Brownian motion gives $B_t/t\to 0$, so $Z_t\to 0$ exponentially and $A_{\infty }<\infty$. Then $Y_s\to 0$ as $s\uparrow A_{\infty }$, and this is its first hit of zero after continuous extension. If $a>0$ the clock diverges by the same strong law. If $a=0$ it diverges by infinite occupation time of one-dimensional Brownian motion on a bounded interval, since $e^{2B_t}$ is bounded below there. Thus the lifetime is infinite when $a\ge 0$. Negative dimensions are interpreted by the same positive-state equation killed at zero; no post-zero Bessel process behavior is asserted.

Let $C_t={\int }_0^t{Z}_s^{2}ds$ and ${\tau }_u=\inf \{t:C_t>u\}$. By the Dambis-Dubins-Schwarz theorem, $W_u=M_{{\tau }_u}$ is Brownian motion. Setting $Y_u=Z_{{\tau }_u}$ and using $du=Z_t^{2}dt$ transforms the decomposition in part b intoComparison with the Bessel process equation $d{Y}_u=d{W}_u+(d-1)(2Y_u{)}^{-1}du$ givesThus an exponential Brownian motion with drift becomes a Bessel process under its quadratic-variation time change; this is an Exponential Brownian-to-Bessel time change.