• Bernoulli
  • Volume 25, Number 3 (2019), 2137-2162.

Integration with respect to the non-commutative fractional Brownian motion

Aurélien Deya and René Schott

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We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian motion in a non-commutative probability setting.

When the Hurst index $H$ of the process is stricly larger than $1/2$, integration can be handled through the so-called Young procedure. The situation where $H=1/2$ corresponds to the specific free case, for which an Itô-type approach is known to be possible.

When $H<1/2$, rough-path-type techniques must come into the picture, which, from a theoretical point of view, involves the use of some a-priori-defined Lévy area process. We show that such an object can indeed be “canonically” constructed for any $H\in(\frac{1}{4},\frac{1}{2})$. Finally, when $H\leq1/4$, we exhibit a similar non-convergence phenomenon as for the non-diagonal entries of the (classical) Lévy area above the standard fractional Brownian motion.

Article information

Bernoulli, Volume 25, Number 3 (2019), 2137-2162.

Received: March 2018
Revised: May 2018
First available in Project Euclid: 12 June 2019

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Zentralblatt MATH identifier

integration theory non-commutative fractional Brownian motion non-commutative stochastic calculus


Deya, Aurélien; Schott, René. Integration with respect to the non-commutative fractional Brownian motion. Bernoulli 25 (2019), no. 3, 2137--2162. doi:10.3150/18-BEJ1048.

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Supplemental materials

  • Supplement to “Integration with respect to the non-commutative fractional Brownian motion”. We provide the technical details of the proof of Proposition 2.8.