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VOL. 51 | 2006 Fractional Brownian fields, duality, and martingales

Editor(s) Evarist Giné, Vladimir Koltchinskii, Wenbo Li, Joel Zinn

## Abstract

In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously. The field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. A mistake common to the existing literature regarding multifractional Brownian motions is pointed out and corrected. The Gaussian field, due to inherited duality'', reveals a new way of constructing martingales associated with the odd and even part of a fractional Brownian motion and therefore of the fractional Brownian motion. The existence of those martingales and their stochastic representations is the first step to the study of natural wavelet expansions associated to those processes in the spirit of our earlier work on a construction of natural wavelets associated to Gaussian-Markov processes.

## Information

Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1129.60036
MathSciNet: MR2387762

Digital Object Identifier: 10.1214/074921706000000770

Subjects:
Primary: 60G15 , 60H10
Secondary: 60G44

Keywords: duality for fractional Brownian motions , fractional Brownian fields , fractional Brownian motion , fundamental martingales