Identification of the Multivariate Fractional Brownian Motion

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.