First-passage time of a Brownian motion: two unexpected journeys

Abstract

The distribution of the first-passage time (FPT)$T_a$ for a Brownian particle with drift $\mu$ subject to hitting an absorber at a level $a>0$ is well-known and given by its density $\gamma (t)=\frac{a}{\sqrt{2\pi t^3}}{e}^{-\frac{(a-\mu t{)}^2}{2t}},t>0$, which is normalized only if $\mu \ge 0$. This article demonstrates the existence of two additional diffusion process categories (one with one parameter and the other with two) that have the same first passage-time distributions when $\mu <0$. For both, we identify the transition densities and thoroughly investigate the processes. A substantial implication is that the first-passage time distribution does not indicate whether the process originates from a drifted Brownian motion or from one of the new processes presented.

final version

References in corpus (4)

• Statistics of first-passage Brownian functionals
• Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon
• On the joint distribution of first-passage time and first-passage area of drifted Brownian motion
• Strongly constrained stochastic processes: the multi-ends Brownian bridge

Cited by in corpus (1)

• Conditioning the tanh-drift process on first-passage times: Exact drifts, bridges, and process equivalences