Drift Identification for L\'{e}vy alpha-Stable Stochastic Systems

TL;DR

This paper introduces a Fourier space method for identifying drift fields in Lévy alpha-stable stochastic systems from time series data, overcoming computational challenges posed by heavy-tailed noise.

Contribution

It develops a novel Fourier-based approach that estimates drift fields by matching characteristic functions, enabling effective system identification with heavy-tailed Lévy noise.

Findings

Successfully learns drift fields in 1D and 2D systems.

Demonstrates qualitative and quantitative accuracy.

Provides a scalable method for heavy-tailed noise systems.

Abstract

This paper focuses on a stochastic system identification problem: given time series observations of a stochastic differential equation (SDE) driven by L\'{e}vy $\alpha$-stable noise, estimate the SDE's drift field. For $\alpha$ in the interval $[1,2)$, the noise is heavy-tailed, leading to computational difficulties for methods that compute transition densities and/or likelihoods in physical space. We propose a Fourier space approach that centers on computing time-dependent characteristic functions, i.e., Fourier transforms of time-dependent densities. Parameterizing the unknown drift field using Fourier series, we formulate a loss consisting of the squared error between predicted and empirical characteristic functions. We minimize this loss with gradients computed via the adjoint method. For a variety of one- and two-dimensional problems, we demonstrate that this method is capable of learning drift fields in qualitative and/or quantitative agreement with ground truth fields.