Integration with Respect to the Hermitian Fractional Brownian Motion

For every $$d\ge 1$$d≥1, we consider the d-dimensional Hermitian fractional Brownian motion (HfBm) that is the process with values in the space of $$(d\times d)$$(d×d)-Hermitian matrices and with upper-diagonal entries given by complex fractional Brownian motions of Hurst index $$H\in (0,1)$$H∈(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 $$H>\frac{1}{3}$$H>13 and identify this interpretation with the rough integral with respect to the $$d^2$$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 $$H\ge \frac{1}{2}$$H≥12, 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.