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This paper focuses on a stochastic system identification problem: given timeseries observations of a stochastic differential equation (SDE) driven byL\'{e}vy $\alpha$-stable noise, estimate the SDE's drift field. For $\alpha$ inthe interval $[1,2)$, the noise is heavy-tailed, leading to computationaldifficulties for methods that compute transition densities and/or likelihoodsin physical space. We propose a Fourier space approach that centers oncomputing time-dependent characteristic functions, i.e., Fourier transforms oftime-dependent densities. Parameterizing the unknown drift field using Fourierseries, we formulate a loss consisting of the squared error between predictedand empirical characteristic functions. We minimize this loss with gradientscomputed via the adjoint method. For a variety of one- and two-dimensionalproblems, we demonstrate that this method is capable of learning drift fieldsin qualitative and/or quantitative agreement with ground truth fields.

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