Identification of the Multivariate Fractional Brownian Motion

TL;DR

This paper develops methods for identifying parameters of multivariate fractional Brownian motion, a complex Gaussian process with multiple Hurst exponents and correlation coefficients, using discrete filtering techniques.

Contribution

It introduces a joint estimation approach for multiple parameters of the multivariate fractional Brownian motion and demonstrates its efficiency through simulations and asymptotic analysis.

Findings

Efficient parameter estimation using discrete filtering.

Asymptotic properties of estimators derived.

Simulation results confirm methodology effectiveness.

Abstract

This paper deals with the identification of the multivariate fractional Brownian motion, a recently developed extension of the fractional Brownian motion to the multivariate case. This process is a $p$-multivariate self-similar Gaussian process parameterized by $p$ different Hurst exponents $H_i$, $p$ scaling coefficients $\sigma_i$ (of each component) and also by $p(p-1)$ coefficients $\rho_{ij},\eta_{ij}$ (for $i,j=1,...,p$ with $j>i$) allowing two components to be more or less strongly correlated and allowing the process to be time reversible or not. We investigate the use of discrete filtering techniques to estimate jointly or separately the different parameters and prove the efficiency of the methodology with a simulation study and the derivation of asymptotic results.