Randomized First Passage Times

TL;DR

This paper investigates the first passage time problem for Brownian motion with a random initial shift, characterizing solutions, proving their uniqueness and existence, and exploring examples with linear boundaries.

Contribution

It introduces a framework for solving inverse first passage time problems for Brownian motion with random initial conditions, including existence and uniqueness results.

Findings

Characterization of solutions for inverse first passage time problems.

Proof of uniqueness and existence of solutions.

Analysis of examples with linear boundary functions.

Abstract

In this article we study a problem related to the first passage and inverse first passage time problems for Brownian motions originally formulated by Jackson, Kreinin and Zhang (2009). Specifically, define $\tau_X = \inf\{t>0:W_t + X \le b(t) \}$ where $W_t$ is a standard Brownian motion, then given a boundary function $b:[0,\infty) \to \RR$ and a target measure $\mu$ on $[0,\infty)$, we seek the random variable $X$ such that the law of $\tau_X$ is given by $\mu$. We characterize the solutions, prove uniqueness and existence and provide several key examples associated with the linear boundary.