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

2009/05/12 by Gerardo Hernández‐del‐Valle, Hernandez-del-Valle, Gerardo
Economics, Econometrics and Finance · Mathematics · #45D05 #45G15 (Secondary) #60J60 (Primary) #60J65 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0905.1971

openalex publication_date 2009/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain explicit solutions for the density φ_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)≥ 0 for all t∈ ℝ⁺. We do so by finding the expected value of some functionals of a 3-dimensional Bessel bridge 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).

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