The Square-Root Wall

Stop a Brownian path the first time it reaches $c$ standard deviations of its own age, $|B_t| = c\sqrt{1+t}$. In the clock $s = \log(1+t)$ the wall becomes the fixed interval $(-c, c)$ for an Ornstein–Uhlenbeck process, and every moment of the wait follows: $\mathbb{E}[(1+\tau_c)^p] = 1/M(-p, \tfrac12, c^2/2)$, with $M$ Kummer’s function. The mean wait $c^2/(1-c^2)$ is finite exactly when $c < 1$, the variance exactly when $c < 0.741964$, and the tail is a power law whose exponent is the first zero of Kummer's function in $p$. A one-sided wall has an exponent below one half at every height, so its mean wait is always infinite. A bracket that widens with $\sqrt{1+t}$, stop at one standard deviation and target at two, wins $20.17$ per cent of the time, not a third, and its exponent $0.3633$ is below one half, so its outcome has no mean. A bridge-corrected log-time simulator confirms every formula whose estimator has a finite variance, within $1.9$ standard errors; a once-a-day check lengthens the mean wait up to $7.7$ times, and the tail exponent recovers only slowly.