Applied Probability Trust (21 November 2009) RANDOMIZED FIRST PASSAGE TIMES

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 τX = inf{t> 0: Wt+X ≤ b(t)} where Wt is a standard Brownian motion, then given a boundary function b: [0,∞) → R and a target measure µ on [0,∞), we seek the random variable X such that the law of τX is given by µ. We characterize the solutions, prove uniqueness and existence and provide several key examples associated with the linear boundary. 1