Mathematical models for complex systems under random fluctuations often certain uncertain parameters. However, quantifying model uncertainty for a stochastic differential equation with an $\alpha$-stable Lévy process is still lacking. Here, we propose an approach to infer all the uncertain non-Gaussian parameters and other system parameters by minimizing the Hellinger distance over the parameter space. The Hellinger distance measures the similarity between an empirical probability density of non-Gaussian observations and a solution (as a probability density) of the associated nonlocal Fokker-Planck equation. Numerical experiments verify that our method is feasible for estimating single and multiple parameters. Meanwhile, we find an optimal estimation interval of the estimated parameters. This method is beneficial for extracting governing dynamical system models under non-Gaussian fluctuations, as in the study of abrupt climate changes in the Dansgaard-Oeschger events.

随机波动下的复杂系统的数学模型通常具有某些不确定的参数。但是，用一个随机变量来量化一类随机微分方程的模型不确定性$\alpha$稳定的列维进程仍然缺乏。在这里，我们提出了一种通过最小化参数空间上的Hellinger距离来推断所有不确定的非高斯参数和其他系统参数的方法。Hellinger距离用于度量非高斯观测的经验概率密度与关联的非局部Fokker-Planck方程的解（作为概率密度）之间的相似性。数值实验验证了我们的方法对于估计单个和多个参数是可行的。同时，我们找到了估计参数的最佳估计间隔。这种方法有助于提取非高斯波动下的控制动力学系统模型，如研究Dansgaard-Oeschger事件中突然的气候变化一样。