Exponential Brownian-to-Bessel time change

Let $Z_t=e^{B_t+at}$ for standard Brownian motion, and set $C_t={\int }_0^t{Z}_s^{2}ds$ and ${\tau }_u=\inf \{t:C_t>u\}$. The Itô formula gives $dZ=ZdB+(a+1/2)Zdt$. For $0\le u<C_{\infty }$, the quadratic variation of the time-changed local martingale is $u$, so the Dambis-Dubins-Schwarz theorem yieldsThe resulting positive process is a Bessel process of dimension $\delta =2a+2$, considered only before its first hit of zero. When $\delta <0$, this describes the killed singular diffusion rather than an everywhere nonnegative squared Bessel process with negative immigration. Its clock lifetime is finite almost surely precisely when $a<0$, equivalently $\delta <2$: negative drift makes the exponential integral finite, whereas the strong law for Brownian motion handles $a>0$ and the Divergence of a driftless Brownian exponential clock handles $a=0$.

The strong law for Brownian motion gives $B_s/s\to 0$. For every $0<\epsilon <\mu$, eventually $e^{(2\mu -2\epsilon )s}\le e^{2B_s+2\mu s}\le e^{(2\mu +2\epsilon )s}$. Integrating these bounds, taking logarithms and dividing by $t$ squeezes the growth rate between $2\mu -2\epsilon$ and $2\mu +2\epsilon$. Let $\epsilon \downarrow 0$. The corresponding inverse clock grows like $(\log t)/(2\mu )$.