Exponential models, brownian motion, and independence

Some examples of steep, reproductive exponential models are considered. These models are shown to possess a τ‐parallel foliation in the terminology of Barndorff‐Nielsen and Blaesild. The independence of certain functions follows directly from the foliation. Suppose X(t) is a Wiener process with drift where X(t) = W(t) + ct , 0 < t < T. Furthermore let Y = max [ X(s), 0 < s < T ]. The joint density of Y and X = X(T) , the end value, is studied within the framework of an exponential model, and it is shown that Y(Y – X ) is independent of X . It is further shown that Y(Y – X) suitably scaled has an exponential distribution. Further examples are considered by randomizing on T .