= Good-lambda bound for Brownian stopping time and maximum
{title2=$\mathbb P(\sqrt\tau>\beta\lambda,B_\tau^*\leq\delta\lambda)\leq\delta^2(\beta^2-1)^{-1}\mathbb P(\sqrt\tau>\lambda)$}
For <Brownian motion> started at zero, condition at time $\lambda^2$. If the stopping time exceeds $\beta^2\lambda^2$ while the running absolute maximum stays below $\delta\lambda$, the path must remain in a fixed interval for another $(\beta^2-1)\lambda^2$ time units. Its expected exit time from this interval is at most $\delta^2\lambda^2$. <Markov's inequality> yields the displayed bound. Integrating it against $d\Phi$ for a function of <moderate growth>, then absorbing the small tail term, proves $\mathbb E\Phi(\sqrt\tau)\leq C_\Phi\mathbb E\Phi(B_\tau^*)$.
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