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Integration with Respect to the Hermitian Fractional Brownian Motion

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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.

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Correspondence to Aurélien Deya.

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Deya, A. Integration with Respect to the Hermitian Fractional Brownian Motion. J Theor Probab 33, 295–318 (2020). https://doi.org/10.1007/s10959-018-0855-8

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Keywords

  • Hermitian fractional Brownian motion
  • Integration theory
  • Pathwise approach
  • Non-commutative stochastic calculus
  • Non-commutative fractional Brownian motion

Mathematics Subject Classification (2010)

  • 15B52
  • 60G22
  • 60H05
  • 46L53