2018/04/13 by Deya, Aurélien
#FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR)
paper · doi:10.48550/arxiv.1804.04917
For every d≥ 1, we consider the d-dimensional Hermitian fractional Brownian motion (HfBm), that is the process with values in the space of (d× d)-Hermitian matrices and with upper-diagonal entries given by complex fractional Brownian motions of Hurst index H∈ (0,1). We follow the approach of [A. Deya and R. Schott: On the rough paths approach to non-commutative stochastic calculus, JFA (2013)] to define a natural integral with respect to the HfBm when H>\frac13, and identify this interpretation with the rough integral with respect to the d2 entries of the matrix. Using this correspondence, we establish a convenient Itô--Stratonovich formula for the Hermitian Brownian motion. Finally, we show that at least when H≥ \frac12, and as the size d of the matrix tends to infinity, the integral with respect to the HfBm converges (in the tracial sense) to the integral with respect to the so-called non-commutative fractional Brownian motion.