The inverse first-passage problem and optimal stopping

TL;DR

This paper establishes a connection between the inverse first-passage problem for Brownian motion and optimal stopping theory, enabling new analytical methods to determine the boundary for a given survival distribution.

Contribution

It demonstrates that the inverse first-passage problem can be solved via an equivalent optimal stopping problem, linking two areas and providing new solution techniques.

Findings

Solution of inverse first-passage problem matches an optimal stopping problem

Methods from optimal stopping theory can be applied to inverse first-passage problems

Analysis of the integral equation for the boundary is facilitated by this connection

Abstract

Given a survival distribution on the positive half-axis and a Brownian motion, a solution of the inverse first-passage problem consists of a boundary so that the first passage time over the boundary has the given distribution. We show that the solution of the inverse first- passage problem coincides with the solution of a related optimal stopping problem. Consequently, methods from optimal stopping theory may be applied in the study of the inverse first-passage problem. We illustrate this with a study of the associated integral equation for the boundary.