Analytic crossing probabilities for certain barriers by Brownian motion

Nabil Kahale

Published 2007-04-21, updated 2008-08-28Version 2

We calculate crossing probabilities and one-sided last exit time densities for a class of moving barriers on an interval $[0,T]$ via Schwartz distributions. We derive crossing probabilities and first hitting time densities for another class of barriers on $[0,T]$ by proving a Schwartz distribution version of the method of images. Analytic expressions for crossing probabilities and related densities are given for new explicit and semi-explicit barriers.