= Exponential Brownian-to-Bessel time change
{c}
Let $Z_t=e^{B_t+at}$ for standard <Brownian motion>, and set $C_t=\int_0^tZ_s^2ds$ and $\tau_u=\inf\{t:C_t>u\}$. The <Itô formula> gives $dZ=Z\,dB+(a+1/2)Z\,dt$. For $0\leq u<C_\infty$, the <quadratic variation> of the time-changed <local martingale> is $u$, so the <Dambis-Dubins-Schwarz theorem> yields
$$
dZ_{\tau_u}=dW_u+\frac{a+1/2}{Z_{\tau_u}}du.
$$
The 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 $a=0$.
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