Unbiased shifts of Brownian motion

Guenter Last, Peter Morters, Hermann Thorisson

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Let B = (B(t) :t∈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∈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 ν on R we construct a stopping time T ≥ 0 with the above properties such that B(T) has distribution ν. 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 R. Another new result is an analogue for diffuse random measures on R of the cycle- stationarity characterisation of Palm versions of stationary simple point processes.
LanguageEnglish
Pages431-463
Number of pages30
JournalAnnals of Probability
Volume42
Issue number2
Early online date18 Jan 2013
DOIs
StatusPublished - Mar 2014

Fingerprint

Brownian motion
Random Measure
Balancing
Stopping Time
Minimality
Local Time
Measurable function
Stationarity
Point Process
Probability Distribution
Moment
Analogue
Cycle
Theorem
Allocation rules
Standards
Probability distribution
Local time
Point process
Stopping time

Keywords

  • Brownian motion
  • Skorokhod embedding
  • local time
  • unbiased shift
  • allocation rule
  • Palm measure
  • random measure

Cite this

Last, G., Morters, P., & Thorisson, H. (2014). Unbiased shifts of Brownian motion. Annals of Probability, 42(2), 431-463. https://doi.org/10.1214/13-AOP832

Unbiased shifts of Brownian motion. / Last, Guenter; Morters, Peter; Thorisson, Hermann.

In: Annals of Probability, Vol. 42, No. 2, 03.2014, p. 431-463.

Research output: Contribution to journalArticle

Last, G, Morters, P & Thorisson, H 2014, 'Unbiased shifts of Brownian motion', Annals of Probability, vol. 42, no. 2, pp. 431-463. https://doi.org/10.1214/13-AOP832
Last G, Morters P, Thorisson H. Unbiased shifts of Brownian motion. Annals of Probability. 2014 Mar;42(2):431-463. https://doi.org/10.1214/13-AOP832
Last, Guenter ; Morters, Peter ; Thorisson, Hermann. / Unbiased shifts of Brownian motion. In: Annals of Probability. 2014 ; Vol. 42, No. 2. pp. 431-463.
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