2018/03/13 by Deya, Aurélien, Schott, René
#FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR)
paper · doi:10.48550/arxiv.1803.04834
We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian 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 \enquotecanonically constructed for any H∈ (\frac14,\frac12). Finally, when H≤ 1/4, we exhibit a similar non-convergence phenomenon as for the non-diagonal entries of the (classical) Lévy area above the standard fractional Brownian.