Is the distribution of the last time Brownian motion crosses a line y=a*x known? (Equivalently, the distribution of the last time a Brownian motion with downwards drift hits 0.) It's not hard to give bounds on it, but it would be nice to have an exact distribution if it's known.
Yes, it is explicitly known, but a bit tedious to write it down. You can find it in section 5.1 (compare also section 4) of
Paavo Salminen. On the First Hitting Time and the Last Exit Time for a Brownian Motion to/from a Moving Boundary. Advances in Applied Probability 20/2, 1998. pp. 411-426.
While I do not know about a free version, you might have institutional access to JSTOR, http://www.jstor.org/stable/1427397