Institute of Mathematical Statistics Collections

Conditional expectations and martingales in the fractional Brownian field

Vladimir Dobrić and Francisco M. Ojeda

Full-text: Open access


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

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

First available in Project Euclid: 2 February 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

fractional Brownian motions fractional Brownian field fundamental martingales

Copyright © 2009, Institute of Mathematical Statistics


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.

Export citation


  • [1] Abramowitz, M. and Stegun, I. A., eds. (1992). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications Inc., New York. Reprint of the 1972 edition.
  • [2] Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions. Encyclopedia of Mathematics and Its Applications 71. Cambridge University Press, Cambridge.
  • [3] Decreusefond, L. (2003). Stochastic integration with respect to fractional Brownian motion. In Theory and Applications of Long-Range Dependence. Birkhäuser Boston, Boston, MA, 203–226.
  • [4] Dobrić, V. and Ojeda, F. M. (2006). Fractional Brownian fields, duality, and martingales. In High Dimensional Probability. IMS Lecture Notes Monogr. Ser. 51 77–95. Inst. Math. Statist., Beachwood, OH.
  • [5] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th ed. Elsevier/Academic Press, Amsterdam. Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).
  • [6] Huang, S. T. and Cambanis, S. (1978). Stochastic and multiple Wiener integrals for Gaussian processes. Ann. Probab. 6(4) 585–614.
  • [7] Jost, C. (2006). Transformation formulas for fractional Brownian motion. Stochastic Process. Appl. 116(10) 1341–1357.
  • [8] Molchan, G. (1969). Gaussian processes with spectra which are asymptotically equivalent to a power of λ. Summaries of papers presented at the sessions of the probability and statistic section of the Moscow Mathematical Society (February-December 1968). Theory Probab. Appl. 14(3) 530–532.
  • [9] Molchan, G. M. and Golosov, J. I. (1969). Gaussian stationary processes with asymptotically a power spectrum. Dokl. Akad. Nauk SSSR 184 546–549.
  • [10] Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4) 571–587.
  • [11] Pipiras, V. and Taqqu, M. S. (2001). Are classes of deterministic integrands for fractional Brownian motion on an interval complete? Bernoulli 7(6) 873–897.
  • [12] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon. Theory and applications, Edited and with a foreword by S. M. Nikolskiĭ, Translated from the 1987 Russian original, Revised by the authors.