Multi-Fractional Brownian Motion: Estimating the Hurst Exponent via Variational Smoothing with Applications in Finance

Abstract

Beginning with the basics of the Wiener process, we consider limitations characterizing the “Brownian approach” in analyzing real phenomena. This leads us to first consider the fractional Brownian motion (fBm)—also discussing the Wood–Chan fast algorithm to generate sample paths—to then focus on multi-fBm and methods to generate its trajectories. This is heavily linked to the Hurst exponent study, which we link to real data, firstly considering an absolute moment method, allowing us to obtain raw estimates, to then consider variational calculus approaches allowing to smooth it. The latter smoothing tool was tested in accuracy on synthetic data, comparing it with the exponential moving average method. Previous analyses and results were exploited to develop a forecasting procedure applied to the real data of foreign exchange rates from the Forex market.