On the inverse first-passage-time problem for a Wiener process

TL;DR

This paper investigates methods to approximate the boundary function in the inverse first-passage-time problem for a Wiener process, analyzing errors and illustrating applications through examples.

Contribution

It introduces two novel approximation methods for the inverse first-passage boundary and studies their errors, with practical examples and potential applications.

Findings

Two approximation methods are proposed and analyzed.

Errors of the methods are quantitatively studied.

Examples demonstrate the methods' effectiveness.

Abstract

The inverse first-passage problem for a Wiener process $(W_t)_{t\ge0}$ seeks to determine a function $b{}:{}\mathbb{R}_+\to\mathbb{R}$ such that \[\tau=\inf\{t>0| W_t\ge b(t)\}\] has a given law. In this paper two methods for approximating the unknown function $b$ are presented. The errors of the two methods are studied. A set of examples illustrates the methods. Possible applications are enlighted.