Maximum before a lower Brownian barrier
= Maximum before a lower Brownian barrier
{title2=$f(x)=\frac b{(b+x)^2}\mathbf1_{\{x>0\}}$}
Stopping Brownian motion at its first crossing below $-b$, the nonnegative local martingale $(W_{t\wedge\tau}+b)/b$ starts at one and ends at zero. The <maximal identity for a continuous nonnegative local martingale tending to zero> gives tail $b/(b+x)$ for the stopped maximum, hence density $b/(b+x)^2$ for $x>0$. There is no atom at zero.