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

Published online by Cambridge University Press:  17 April 2009

Shey Shiung Sheu
Shey Shiung Sheu
Affiliation:
Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China.
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Abstract

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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].

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 3 , December 1986 , pp. 427 - 431
Copyright
Copyright © Australian Mathematical Society 1986

References

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