The inverse first-passage time problem as hydrodynamic limit of a particle system

Abstract

Given a distribution on the positive real line, the two-sided inverse first-passage time problem for Brownian motion asks for a function, such that the first-passage time of this function by a reflected Brownian motion has the given distribution. We present a particle system without branching but with selection at timepoints depending on the given distribution on the positive real line, whose hydrodynamic limit is the distribution of a Brownian motion conditioned to not have passed the solution of the inverse first-passage time problem. As application we extract a simple Monte-Carlo method to simulate these solutions.

An elementary approach to the inverse first-passage-time problem for soft-killed Brownian motion

“…Heuristically, q(n) k is an empirical version of q (n) (kδ (n) ) from (2.6). A proof for the validity of this choice, namely that q(n) k → q (n) (kδ (n) ) almost surely as N → ∞, can be found in [18,Theorem 3.3.2].…”