= Brownian symmetric interval-exit moments
{c}
{title2=$\mathbb E\tau_x=x^2,\quad\mathbb E\tau_x^2=5x^4/3$}
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 $2x^4/3$.
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