A randomized first-passage problem for drifted Brownian motion subject to hold and jump from a boundary

TL;DR

This paper investigates an inverse first-passage-time problem for a drifted Brownian motion with hold and jump dynamics at a boundary, aiming to determine the holding and jump distributions that produce a specified first-passage time distribution.

Contribution

It introduces a novel inverse problem framework for drifted Brownian motion with boundary hold and jump conditions, providing methods to identify distributions that match a target first-passage time.

Findings

Derived conditions for the existence of solutions.

Provided explicit formulas for the distributions.

Demonstrated applicability through examples.

Abstract

We study an inverse first-passage-time problem for Wiener process $X(t)$ subject to hold and jump from a boundary $c.$ Let be given a threshold $S>X(0) \ge c,$ and a distribution function $F$ on $[0, + \infty ).$ The problem consists in finding the distribution of the holding time at $c$ and the distribution of jumps from $c,$ so that the first-passage time of $X(t)$ through $S$ has distribution $F.$