Tempered fractional Brownian motion: wavelet estimation, modeling and testing

TL;DR

This paper introduces a wavelet-based estimation method and a testing procedure for tempered fractional Brownian motion (tfBm), demonstrating its effectiveness in modeling geophysical data more accurately than traditional fBm.

Contribution

It presents the first wavelet-based estimation and testing methods for tfBm, a model capturing semi-long range dependence and non-scaling behaviors.

Findings

Wavelet estimator for tfBm is mathematically and computationally validated.

The test effectively distinguishes between fBm and tfBm.

Application to geophysical data shows tfBm fits better than fBm.

Abstract

The Davenport spectrum is a modification of the classical Kolmogorov spectrum for the inertial range of turbulence that accounts for non-scaling low frequency behavior. Like the classical fractional Brownian motion vis-\`a-vis the Kolmogorov spectrum, tempered fractional Brownian motion (tfBm) is a canonical model that displays the Davenport spectrum. The autocorrelation of the increments of tfBm displays semi-long range dependence (hyperbolic and quasi-exponential decays over moderate and large scales, respectively), a phenomenon that has been observed in wide a range of applications from wind speeds to geophysics to finance. In this paper, we use wavelets to construct the first estimation method for tfBm and a simple and computationally efficient test for fBm vs tfBm alternatives. The properties of the wavelet estimator and test are mathematically and computationally established. An application of the methodology to the analysis of geophysical flow data shows that tfBm provides a much closer fit than fBm.