A maximum likelihood estimation of Lévy-driven stochastic systems for univariate and multivariate time series of observations
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单变量与多变量观测时序中 Lévy 驱动随机系统的最大似然估计
- 研究问题
- 如何从含有不同大小与频率跳跃的观测时序重建 Lévy 驱动随机系统，并同时估计系统及噪声参数？
- 方法与思路
- 作者提出最大似然估计程序，处理单变量和多变量观测，并以模拟数据及覆盖末次冰期的冰芯数据展示应用。
- 材料陈述的贡献
- 摘要称该程序可完整重建系统、降低数据需求且便于多变量实现，并在冰芯应用中得到此前研究未发现的气候动态信息。
- 阅读关注点
- 应核对似然的构造与近似、观测采样要求、可重建的噪声参数范围，以及数据需求比较和冰芯结果的验证方式。
- 阅读边界
- 这是 arXiv 预印本，同行评审及接收未核实；摘要未提供具体条件或结果，完整重建、较少数据及首创性仍需核对。
原文摘要
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 Lévy type. Reconstruction of Lévy-driven dynamical systems, however, has been a major challenge. The literature on the reconstruction of Lévy-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.