Integration with Respect to the Hermitian Fractional Brownian Motion

Abstract

For every $d\ge 1$, we consider the d-dimensional Hermitian fractional Brownian motion (HfBm) that is the process with values in the space of $(d\times d)$-Hermitian matrices and with upper-diagonal entries given by complex fractional Brownian motions of Hurst index $H\in (0,1)$. We follow the approach of Deya and Schott [J Funct Anal 265(4):594–628, 2013] to define a natural integral with respect to the HfBm when $H>\frac{1}{3}$ and identify this interpretation with the rough integral with respect to the $d^2$ 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 $H\ge \frac{1}{2}$, 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.