Wavelet estimation for operator fractional Brownian motion

TL;DR

This paper introduces a wavelet-based method to estimate the Hurst matrix of operator fractional Brownian motion, overcoming challenges of entry-wise analysis by focusing on eigenstructure evolution, with proven consistency and normality.

Contribution

It develops a novel wavelet analysis approach for OFBM, providing consistent, asymptotically normal estimators of the Hurst eigenvalues and eigenstructure.

Findings

Estimators are consistent and asymptotically normal.

Simulation shows good finite-sample performance.

Eigenstructure-based analysis improves estimation accuracy.

Abstract

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. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. The proposed approach consists of considering the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the coordinate system itself under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.