Method of Filtration in first passage time problems

TL;DR

This paper introduces a filtration method to derive the first passage time distribution for finite intervals with two absorbing boundaries from the semi-infinite case, generalizing classical solutions like eigenfunction expansion and image methods.

Contribution

The paper presents a novel filtration approach that constructs solutions for finite interval first passage problems from semi-infinite cases, extending the method of images to processes like Ornstein-Uhlenbeck.

Findings

Provides a filtration method to derive finite interval distributions

Generalizes the method of images for complex stochastic processes

Offers two solution forms: eigenfunction expansion and image-like

Abstract

Statistics of stochastic processes are crucially influenced by the boundary conditions. In one spatial dimension, for example, the first passage time distribution in semi-infinite space (one absorbing boundary) is markedly different from that in a finite interval with two absorbing boundaries. Here, we propose a method, which we refer to as a method of filtration, that allows us to construct the latter from only the knowledge of the former. We demonstrate that our method yields two solution forms, a method of eigenfunction expansion-like form and a method of image-like form. In particular, we argue that the latter solution form is a generalization of the method of image applicable to a stochastic process for which the method of image generally does not work, e.g., the Ornstein-Uhlenbeck process.