Drifted Brownian first-passage density
= Drifted Brownian first-passage density
{title2=$h_c(t)=ae^{-(a-ct)^2/(2t)}/\sqrt{2\pi t^3}$}
For a positive level $a$, this density describes the <first-passage time> of $B_t+ct$. The <Girsanov theorem> multiplies the zero-drift density by the stopped likelihood $e^{ca-c^2t/2}$. Its total mass is one for nonnegative drift and $e^{2ac}$ for negative drift; the latter has remaining mass at infinite hitting time.