Unbiased shifts of Brownian motion

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 $T\ge 0$ 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.

Published in at http://dx.doi.org/10.1214/13-AOP832 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• The Skorokhod embedding problem and its offspring
• Extra heads and invariant allocations
• Invariant transports of stationary random measures and mass-stationarity

Cited by in corpus (4)

• On a solution to the Monge transport problem on the real line arising from the strictly concave case
• Transporting random measures on the line and embedding excursions into Brownian motion
• Stable transports between stationary random measures
• Optimal embeddings by unbiased shifts of Brownian motion