## 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**

Received: March 2018

Revised: May 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.3150/18-BEJ1048

**Mathematical Reviews number (MathSciNet)**

MR3961243

**Zentralblatt MATH identifier**

07066252

**Keywords**

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

#### 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

#### 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.Digital Object Identifier: doi:10.3150/18-BEJ1048SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.