Representing a distribution by stopping a Brownian Motion: Root's construction

A closed subset c of [0,∞]×[−∞,∞] is called a barrier if (i) (∞,x) ∈ C, ∀̱x, (ii) (t, ±∞) ∈ C, ∀̱t, (iii) (t, x) ∈ C implies (s, x) ∈ C, ∀̱S ≥ t. Given a Brownian motion (B (t)) Starting at the origin and a barrier C, let τ(C) be inf{t: (t, B (t)) ∈ C}. A random variable x (or a distribution F) is called achievable if there exists a barrier C so that B (τ(C)) is distributed as x (F). In this paper we shall show that if x is bounded above or below with finite mean or if x has zero mean and E (|x| log+ |x|) < ∞ then x is achievable. This result gives a partial answer to a problem raised by Loynes [7].