### Abstract

Original language | English |
---|---|

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 |

### Fingerprint

### Keywords

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

### Cite this

*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.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 42, no. 2, pp. 431-463. https://doi.org/10.1214/13-AOP832

}

TY - JOUR

T1 - Unbiased shifts of Brownian motion

AU - Last, Guenter

AU - Morters, Peter

AU - Thorisson, Hermann

PY - 2014/3

Y1 - 2014/3

N2 - 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.

AB - 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.

KW - Brownian motion

KW - Skorokhod embedding

KW - local time

KW - unbiased shift

KW - allocation rule

KW - Palm measure

KW - random measure

UR - http://www.scopus.com/inward/record.url?scp=84894559597&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1214/13-AOP832

U2 - 10.1214/13-AOP832

DO - 10.1214/13-AOP832

M3 - Article

VL - 42

SP - 431

EP - 463

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 2

ER -