Distribution of last time Brownian motion crosses a line

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.

• That question discusses the first hitting time, whereas I want the last. Is there an obvious connection that I'm missing? – Elena Yudovina Oct 3 '13 at 21:09
• A simple description of that distribution is given in mathoverflow.net/questions/222705/… – esg Apr 28 '17 at 16:07

1 Answer

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

• cant it be derived using the analysis done in Billingsley's book convergence of probability measures? he does something very similar for the last time to hit $0$. – user24367 Oct 31 '13 at 7:31