Randomized First Passage Times

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.

arXiv comment24 pages, 2 figures