Bulletin of the Australian Mathematical Society

Table of Contents - 1986 - Volume 34, Issue 03  

Research Article

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

Shey Shiung Sheua1

a1 Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China.

Abstract

A closed subset c of [0,∞]×[−∞,∞] is called a barrier if

(i) (∞,x) xs2208 C, x,

(ii) (t, ±∞) xs2208 C, t,

(iii) (t, x) xs2208 C implies (s, x) xs2208 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)) xs2208 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].

(Received January 28 1986)