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March 2014 Unbiased shifts of Brownian motion
Günter Last, Peter Mörters, Hermann Thorisson
Ann. Probab. 42(2): 431-463 (March 2014). DOI: 10.1214/13-AOP832

Abstract

Let $B=(B_{t})_{t\in\mathbb{R}}$ be a two-sided standard Brownian motion. An unbiased shift of $B$ is a random time $T$, which is a measurable function of $B$, such that $(B_{T+t}-B_{T})_{t\in\mathbb{R}}$ is a Brownian motion independent of $B_{T}$. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of $B$. For any probability distribution $\nu$ on $\mathbb{R}$ we construct a stopping time $T\ge0$ with the above properties such that $B_{T}$ has distribution $\nu$. We also study moment and minimality properties of unbiased shifts. A crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on $\mathbb{R}$. Another new result is an analogue for diffuse random measures on $\mathbb{R}$ of the cycle-stationarity characterisation of Palm versions of stationary simple point processes.

Citation

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Günter Last. Peter Mörters. Hermann Thorisson. "Unbiased shifts of Brownian motion." Ann. Probab. 42 (2) 431 - 463, March 2014. https://doi.org/10.1214/13-AOP832

Information

Published: March 2014
First available in Project Euclid: 24 February 2014

zbMATH: 1295.60093
MathSciNet: MR3178463
Digital Object Identifier: 10.1214/13-AOP832

Subjects:
Primary: 60G55, 60G57, 60J65

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 2 • March 2014
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