Fractional Brownian fields, duality, and martingales

Details Fractional Brownian fields, duality, and martingales Fractional Brownian fields, duality, and martingales

2006-12-22

arxiv.org/abs/math/0612694v1

IMS Lecture Notes Monograph Series 2006, Vol. 51, 77-95

Mathematics

Probability

Published at http://dx.doi.org/10.1214/074921706000000770 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/p

Scientific paper

10.1214/074921706000000770

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.

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