Regularized Learning for Fractional Brownian Motion via Path Signatures

TL;DR

This paper investigates the use of signature-based Lasso regression for modeling fractional Brownian motion, demonstrating its effectiveness in handling irregular paths and outperforming traditional methods through theoretical analysis and simulations.

Contribution

It introduces a novel application of signature Lasso regression to fractional Brownian motion, including theoretical consistency results for different Hurst parameters.

Findings

Signature Lasso outperforms traditional regression on synthetic data.

Theoretical bounds established for signature moments for different Hurst parameters.

Simulation results confirm improved performance on real-world datasets.

Abstract

Fractional Brownian motion (fBm) extends classical Brownian motion by introducing dependence between increments, governed by the Hurst parameter $H\in (0,1)$. Unlike traditional Brownian motion, the increments of an fBm are not independent. Paths generated by fractional Brownian motions can exhibit significant irregularity, particularly when the Hurst parameter is small. As a result, classical regression methods may not perform effectively. Signatures, defined as iterated path integrals of continuous and discrete-time processes, offer a universal nonlinearity property that simplifies the challenge of feature selection in time series data analysis by effectively linearizing it. Consequently, we employ Lasso regression techniques for regularization when handling irregular data. To evaluate the performance of signature Lasso on fractional Brownian motion (fBM), we study its consistency when the Hurst parameter $ H \ne \frac{1}{2} $. This involves deriving bounds on the first and second moments of the signature. For the case $ H > \frac{1}{2} $, we use the signature defined in the Young sense, while for $ H < \frac{1}{2} $, we use the Stratonovich interpretation. Simulation results indicate that signature Lasso can outperform traditional regression methods for synthetic data as well as for real-world datasets.