On Schroedinger's equation, 3-dimensional bessel bridges, and passage time problems

TL;DR

This paper derives explicit formulas for the first-passage time density of a Brownian motion to a smooth, convex moving boundary, using connections to 3-dimensional Bessel bridges and Schrödinger's equation.

Contribution

It provides a novel explicit solution for passage time densities involving Bessel bridges and Schrödinger's equation, extending previous theoretical frameworks.

Findings

Explicit density formulas for first-passage times to convex boundaries.

Connection between Bessel bridges and Schrödinger's equation with time-dependent potential.

Application of stochastic process functionals to solve passage time problems.

Abstract

We obtain explicit solutions for the density $\varphi_T$ of the first-time $T$ that a one-dimensional Brownian process $B$ reaches the twice, continuously differentiable moving boundary $f$ and such that $f''(t)\geq 0$ for all $t\in \mathbb{R}^+$. We do so by finding the expected value of some functionals of a 3-dimensional Bessel bridge $\tilde{X}$ and exploiting its relationship with first-passage time problems as pointed out by Kardaras (2007). It turns out that this problem is related to Schr\"odinger's equation with time-dependent linear potential, see Feng (2001).