Maximum before a lower Brownian barrier ($f(x)=\frac{b}{(b+x{)}^2}{1}_{\{x>0\}}$)

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.