Drift Identification for Lévy alpha-Stable Stochastic Systems
This addresses a computational challenge in system identification for heavy-tailed stochastic systems, but it is incremental as it adapts existing Fourier and adjoint methods to a specific noise type.
The paper tackles the problem of estimating the drift field in stochastic differential equations driven by heavy-tailed Lévy alpha-stable noise, proposing a Fourier space method that learns drift fields in agreement with ground truth for one- and two-dimensional problems.
This paper focuses on a stochastic system identification problem: given time series observations of a stochastic differential equation (SDE) driven by Lévy $α$-stable noise, estimate the SDE's drift field. For $α$ 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.