Solution
= Solution
The <Cameron-Martin theorem for a linear drift> changes the density of Brownian paths through time $t$ by
$$
\exp\left(bB_t-\frac12b^2t\right).
$$
At the driftless hitting time $\tau_{a,0}=t$, the endpoint is $B_t=a$. Multiplying its given density by the likelihood $e^{ab-b^2t/2}$ therefore yields
$$
a(2\pi t^3)^{-1/2}
\exp\left(-\frac{a^2}{2t}+ab-\frac12b^2t\right)
=a(2\pi t^3)^{-1/2}
\exp\left(-\frac{(a-bt)^2}{2t}\right).
$$