Basic properties of the Multivariate Fractional Brownian Motion

TL;DR

This paper reviews and extends key properties of multivariate fractional Brownian motion, including covariance, spectral analysis, and simulation methods, providing a comprehensive understanding and practical algorithms for this stochastic process.

Contribution

It introduces a new characterization of mfBm via covariance, analyzes its increments, and presents an exact simulation algorithm, advancing both theoretical understanding and practical implementation.

Findings

Covariance characterization of mfBm

Spectral analysis of increments

Algorithm for perfect simulation of mfBm

Abstract

This paper reviews and extends some recent results on the multivariate fractional Brownian motion (mfBm) and its increment process. A characterization of the mfBm through its covariance function is obtained. Similarly, the correlation and spectral analyses of the increments are investigated. On the other hand we show that (almost) all mfBm's may be reached as the limit of partial sums of (super)linear processes. Finally, an algorithm to perfectly simulate the mfBm is presented and illustrated by some simulations.