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Conditional expectations and martingales in the fractional Brownian field

Vladimir Dobrić and Francisco M. Ojeda

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Abstract

Conditional expectations of a fractional Brownian motion with Hurst index H respect to the filtration of a fractional Brownian motion with Hurst index H, both contained in the fractional Brownian field, are studied. A stochastic integral representation of those processes is constructed from the covariance structure of the underlying fractional Brownian field. As processes, the conditional expectations contain martingale components and for dual pairs of Hurst indices the processes become pure martingales which, up to a multiplicative constant, coincide with the fundamental martingales of fractional Brownian motions.

Chapter information

Source
Christian Houdré, Vladimir Koltchinskii, David M. Mason and Magda Peligrad, eds., High Dimensional Probability V: The Luminy Volume (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2009), 224-238

Dates
First available in Project Euclid: 2 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1265119271

Digital Object Identifier
doi:10.1214/09-IMSCOLL515

Mathematical Reviews number (MathSciNet)
MR2797950

Zentralblatt MATH identifier
1243.60037

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G18: Self-similar processes 60G44: Martingales with continuous parameter

Keywords
fractional Brownian motions fractional Brownian field fundamental martingales

Rights
Copyright © 2009, Institute of Mathematical Statistics

Citation

Dobrić, Vladimir; Ojeda, Francisco M. Conditional expectations and martingales in the fractional Brownian field. High Dimensional Probability V: The Luminy Volume, 224--238, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2009. doi:10.1214/09-IMSCOLL515. https://projecteuclid.org/euclid.imsc/1265119271


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