Drift Identification for Lévy alpha-Stable Stochastic Systems
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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.
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