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Fractional Brownian Fields, Duality, and Martingales

Vladimir Dobrić and Francisco M. Ojeda
Lecture Notes-Monograph Series
Vol. 51, High Dimensional Probability (2006), pp. 77-95
Stable URL: http://www.jstor.org/stable/4356434
Page Count: 19
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Fractional Brownian Fields, Duality, and Martingales
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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.

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