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

Abstract

We obtain explicit solutions for the density ${\phi }_T$ of the first-time $T$ that a one-dimensional Brownian process $B$ reaches the twice, continuously differentiable moving boundary $f$ and such that $f^{\prime \prime }(t)\ge 0$ 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 Schrödinger's equation with time-dependent linear potential, see Feng (2001).