### 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.

Original language | English |
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Pages (from-to) | 431-463 |

Number of pages | 30 |

Journal | Annals of Probability |

Volume | 42 |

Issue number | 2 |

Early online date | 18 Jan 2013 |

DOIs | |

Publication status | Published - Mar 2014 |

### Keywords

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

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## 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