Non-Linear Wavelet Regression and Branch & Bound Optimization for the Full Identification of Bivariate Operator Fractional Brownian Motion

TL;DR

This paper introduces a novel method combining non-linear wavelet regression and Branch & Bound optimization for the comprehensive joint estimation of all parameters in bivariate Operator Fractional Brownian Motion, a multivariate self-similarity model.

Contribution

It presents the first full identification approach for bivariate OfBm, addressing estimation challenges with a new formulation and rigorous performance analysis.

Findings

Estimation method is consistent and asymptotically normal.

Numerical experiments validate the effectiveness of the approach.

Parameter impacts on estimation accuracy and computational costs are analyzed.

Abstract

Self-similarity is widely considered the reference framework for modeling the scaling properties of real-world data. However, most theoretical studies and their practical use have remained univariate. Operator Fractional Brownian Motion (OfBm) was recently proposed as a multivariate model for self-similarity. Yet it has remained seldom used in applications because of serious issues that appear in the joint estimation of its numerous parameters. While the univariate fractional Brownian motion requires the estimation of two parameters only, its mere bivariate extension already involves 7 parameters which are very different in nature. The present contribution proposes a method for the full identification of bivariate OfBm (i.e., the joint estimation of all parameters) through an original formulation as a non-linear wavelet regression coupled with a custom-made Branch & Bound numerical scheme. The estimation performance (consistency and asymptotic normality) is mathematically established and numerically assessed by means of Monte Carlo experiments. The impact of the parameters defining OfBm on the estimation performance as well as the associated computational costs are also thoroughly investigated.