Literature is full of inference techniques developed to estimate the parameters of stochastic dynamical systems driven by the well-known Brownian noise. Such diffusion models are often inappropriate models to properly describe the dynamics reflected in many real-world data which are dominated by jump discontinuities of various sizes and frequencies. To account for the presence of jumps, jump-diffusion models are introduced and some inference techniques are developed. Jump-diffusion models are also inadequate models since they fail to reflect the frequent occurrence as well as the continuous spectrum of natural jumps. It is, therefore, crucial to depart from the classical stochastic systems like diffusion and jump-diffusion models and resort to stochastic systems where the regime of stochasticity is governed by the stochastic fluctuations of L\'evy type. Reconstruction of L\'evy-driven dynamical systems, however, has been a major challenge. The literature on the reconstruction of L\'evy-driven systems is rather poor: there are few reconstruction algorithms developed which suffer from one or several problems such as being data-hungry, failing to provide a full reconstruction of noise parameters, tackling only some specific systems, failing to cope with multivariate data in practice, lacking proper validation mechanisms, and many more. This letter introduces a maximum likelihood estimation procedure which grants a full reconstruction of the system, requires less data, and its implementation for multivariate data is quite straightforward. To the best of our knowledge this contribution is the first to tackle all the mentioned shortcomings. We apply our algorithm to simulated data as well as an ice-core dataset spanning the last glaciation. In particular, we find new insights about the dynamics of the climate in the curse of the last glaciation which was not found in previous studies.

翻译：文献中充斥着针对由著名布朗噪声驱动的随机系统参数估计的推理技术。此类扩散模型往往不适用于准确描述许多真实世界数据所反映的动态特性，这些数据以大小和频率各异的跳跃间断为主导。为考虑跳跃的存在，引入了跳跃扩散模型并开发了部分推理技术。然而，跳跃扩散模型仍非充分模型，因其无法反映频繁发生的跳跃以及自然跳跃的连续谱特性。因此，关键在于摆脱经典的随机系统（如扩散模型和跳跃扩散模型），转向以Lévy型随机波动主导随机性机制的随机系统。然而，Lévy驱动动力系统的重构一直是一项重大挑战。关于Lévy驱动系统重构的文献相对匮乏：已开发的少数重构算法存在一个或多个问题，例如数据需求量大、无法实现噪声参数的完全重构、仅能处理特定系统、实践中无法应对多变量数据、缺乏有效的验证机制等。本文提出一种最大似然估计方法，该方法能够实现系统的完全重构，所需数据量更少，且对多变量数据的实现相当直接。据我们所知，本研究首次解决了上述所有不足。我们将该算法应用于模拟数据以及覆盖末次冰期的冰芯数据集。特别地，我们发现了关于末次冰期气候动态的新见解，这在以往研究中未曾发现。