Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 4 / d / Solution

The lifetime of the time-changed process isBy the strong law for Brownian motion, $B_t/t\to 0$ almost surely. If $a<0$, the exponent is eventually at most $at<0$, so the integral is finite. If $a>0$, it is eventually positive and grows linearly; if $a=0$, the recurrence of one-dimensional Brownian motion makes $B$ spend infinite total time in, for example, $[-1,1]$, so the integral is infinite. Consequently