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.

翻译：本文侧重于一个随机系统识别问题:根据时间序列对L\'{{e}vy $\alpha$- sable 噪声驱动的随机差异方程式(SDE)进行的时间序列观测,估计SDE的漂流场。对于间隔为$\alpha$($11,2美元),噪音是重尾的,导致计算转换密度和/或物理空间可能性的方法的计算困难。我们提议了一个四维空间方法,该方法以计算取决于时间的特性函数为中心,即根据时间的密度的Fourier变异为Fourier。用四维序列参数参数对未知的漂流场进行参数测量,我们用预测的特性函数与实验性特性函数之间的平方差来计算损失。我们用联合方法计算出的梯度来尽量减少这一损失。对于各种一维和二维问题,我们证明这种方法能够学习与地面真理字段在质和/或定量协议中的漂移场。