Schwarz, W. & Miller, J. O. (2010). Locking the Wiener process to its level-crossing time. Communications in Statistics -­ Theory and Methods, 39 (2), 372-381.

We consider the specific transformation of a Wiener process {X(t),  t ³ 0} in the presence of an absorbing barrier a that results when this process is "time-locked" with respect to its first passage time Ta through a criterion level a, and the evolution of X(t) is considered backwards (retrospectively) from Ta. Formally, we study the random variables defined by Y(t) º X(Ta-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.