Locking the Wiener Process to its Level-Crossing Time

We consider the specific transformation of a Wiener process {X(t), t >= 0} in the presence of an absorbing barrier a that results when this process is "time-locked" with respect to its first passage time T(a) through a criterion level a, and the evolution of X(t) is considered backwards ( retrospectively) from T(a). Formally, we study the random variables defined by Y(t) = X(T(a) - t) and derive explicit results for their density and mean, and also for their asymptotic forms. We discuss how our results can aid interpretations of time series "response-locked" to their times of crossing a criterion level.