Conditioning a diffusion at first-passage and last-exit times, and a mirage arising in drug therapy for HIV

Let X(t), t > or = 0, be a real-valued diffusion process, not necessarily time-homogeneous. For a fixed z, let Tz be the first-passage time to z; and, under the assumption that z is a transient state, let Vz be the last-exit time from z. The main results of this work are a description of the structure of X(t), t > or = 0, upon conditioning by the values of Tz and Vz, and some of the characteristics of the conditioned process. The results are illustrated by application to Brownian motion with constant drift and also to the Ornstein-Uhlenbeck process augmented by a linear term representing a constant drift (a nonhomogeneous diffusion). The problem was motivated by the need to explain a possible mathematical mirage in the testing of antiviral drugs for HIV infection.