Integration with Respect to the Hermitian Fractional Brownian Motion

Aurélien Deya ()
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Aurélien Deya: University of Lorraine

Journal of Theoretical Probability, 2020, vol. 33, issue 1, 295-318

Abstract: Abstract 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.

Keywords: Hermitian fractional Brownian motion; Integration theory; Pathwise approach; Non-commutative stochastic calculus; Non-commutative fractional Brownian motion; 15B52; 60G22; 60H05; 46L53 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-018-0855-8

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