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

• URL: http://arxiv.org/abs/2212.03317v1
• Date: Tue, 6 Dec 2022 20:40:27 GMT
• Title: Drift Identification for L\'{e}vy alpha-Stable Stochastic Systems
• Authors: Harish S. Bhat
• Abstract summary: Given time series observations of a differential equation, estimate the SDE's drift field. For $alpha$ in the interval $[1,2)$, the noise is heavy-tailed. We propose a space approach that centers on computing time-dependent characteristic functions.
• Score: 2.28438857884398
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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