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Fractional Brownian fields, duality, and martingales. (English) Zbl 1129.60036

Giné, Evarist (ed.) et al., High dimensional probability. Proceedings of the fourth international conference. Many papers based on the presentations at the conference, Santa Fe, NM, USA, June 20–24, 2005. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-67-6/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 51, 77-95 (2006).
The article is devoted to the integral representation of the fractional Brownian motion. The main result is the construction of the centered Gaussian random field \(\{Z_H(t)\}_{t\in\mathbb R, H\in (0;1)}\) with the covariance \[ EZ_H(t)Z_{\acute{H}}(s)=a{H,\acute{H}}\tfrac{1}{2}(| t| ^{H+\acute{H}}+| s| ^{H+\acute{H}}-| t-s| ^{H+\acute{H}}). \] The authors consider this field as a family of properly correlated fractional Brownian motions. Using it the authors present the Gaussian martingales, which generate the same filtration as an even and odd parts of the fractional Brownian motion.
For the entire collection see [Zbl 1113.60009].

MSC:

60G15 Gaussian processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter
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