Brownian exit from an interval

For one-dimensional Brownian motion started at zero and $T={\tau }_b\wedge {\tau }_{-a}$ with $a,b>0$,Optional stopping of $B_t$ and $B_t^2-t$ proves the two identities.

The exit time conditioned on leaving through the upper endpoint satisfiesOptional stopping of the cubic martingale $B_t^3-3t{B}_t$, together with the lower moments of the interval exit, gives the formula.