Brownian symmetric interval-exit moments ($\mathbb{E}{\tau }_x=x^2,\mathbb{E}{\tau }_x^2=5x^4/3$)

For standard Brownian motion started at zero, let ${\tau }_x$ be its first exit from $(-x,x)$, with $x>0$. Bounded-time stopping of $B_t^2-t$ first proves integrability of the exit time and then gives its mean $x^2$. Stopping the quartic Hermite polynomial martingale gives a uniform bound on $\mathbb{E}(t\wedge {\tau }_x{)}^2$, proving second-moment integrability before passing to the limit. The resulting second moment is $5x^4/3$ and the variance is $2x^4/3$.