Integral representations and properties of operator fractional Brownian motions

TL;DR

This paper characterizes operator fractional Brownian motions using integral representations in spectral and time domains, clarifying their covariance structure, time-reversibility, and spectral density properties.

Contribution

It provides the first comprehensive integral representations of OFBMs in spectral and time domains, addressing covariance and time-reversibility issues.

Findings

Integral representations of OFBMs in spectral and time domains.

Necessary and sufficient conditions for time-reversibility.

The spectral density of stationary increments exhibits a rigid structure called the dichotomy principle.

Abstract

Operator fractional Brownian motions (OFBMs) are (i) Gaussian, (ii) operator self-similar and (iii) stationary increment processes. They are the natural multivariate generalizations of the well-studied fractional Brownian motions. Because of the possible lack of time-reversibility, the defining properties (i)--(iii) do not, in general, characterize the covariance structure of OFBMs. To circumvent this problem, the class of OFBMs is characterized here by means of their integral representations in the spectral and time domains. For the spectral domain representations, this involves showing how the operator self-similarity shapes the spectral density in the general representation of stationary increment processes. The time domain representations are derived by using primary matrix functions and taking the Fourier transforms of the deterministic spectral domain kernels. Necessary and sufficient conditions for OFBMs to be time-reversible are established in terms of their spectral and time domain representations. It is also shown that the spectral density of the stationary increments of an OFBM has a rigid structure, here called the dichotomy principle. The notion of operator Brownian motions is also explored.