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

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

2009-05-12

arxiv.org/abs/0905.1971v1

Mathematics

Probability

Scientific paper

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).

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