论文标题
LévyAlpha稳定随机系统的漂移识别
Drift Identification for Lévy alpha-Stable Stochastic Systems
论文作者
论文摘要
本文重点介绍了随机系统识别问题：给定时间序列观察，对由Lévy$α$稳定噪声驱动的随机微分方程（SDE），估计SDE的漂移场。对于间隔$ [1,2）$的$α$，噪声是重尾，导致计算困难，用于计算物理空间中过渡密度和/或可能性的方法。我们提出了一种傅立叶空间方法，该方法集中于计算时间相关的特征函数，即时间相关密度的傅立叶变换。使用傅立叶级数对未知漂移场进行参数化，我们制定了由预测和经验特征函数之间的平方误差组成的损耗。我们通过通过伴随方法计算的梯度最大程度地减少了这一损失。对于各种一维问题，我们证明了该方法能够与地面真实领域的定性和/或定量一致学习漂移领域。
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.
