Locking the Wiener Process to its Level-Crossing Time

Abstract

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.

Keywords:

• First hitting model
• Inverse Gaussian distribution
• Response-locked time series
• Wiener diffusion process

Mathematics Subject Classification:

• Primary 60J70
• Secondary 91E30
• 92C05

Notes

¹We thank an anonymous reviewer for suggesting this more abstract and compact derivation.