## Bernoulli

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

### Integration with respect to the non-commutative fractional Brownian motion

#### Abstract

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

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

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

https://projecteuclid.org/euclid.bj/1560326440

Digital Object Identifier
doi:10.3150/18-BEJ1048

Mathematical Reviews number (MathSciNet)
MR3961243

Zentralblatt MATH identifier
07066252

#### Citation

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. https://projecteuclid.org/euclid.bj/1560326440

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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.