Wavelet estimation of operator fractional Brownian motions

Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. Despite the theoretical relevance of OFBMs, the difficulties associated with the estimation of the generally non-diagonal matrix Hurst parameter have effectively prevented its use in applications. This paper develops the wavelet analysis of OFBMs, as well as a new estimator for the Hurst matrix of bivariate OFBMs. Our approach relies on an original change of perspective: instead of considering the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales, it draws upon the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst index in each coordinate, and of the coordinate system itself under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.