Lévy processes, known for their ability to model complex dynamics with skewness, heavy tails, and discontinuities, play a critical role in stochastic modeling across various domains. However, inference for most Lévy processes, whether in parametric or non-parametric settings, remains a significant challenge. In this work, we present a novel Bayesian non-parametric inference framework for inferring the Lévy measures of subordinators and normal variance-mean (NVM) processes within a linear state space model. A flexible random measure, the Independent Gamma-scaled Dirichlet Process (IGSDP), is introduced, for which the well-known Gamma process is a special case, leading to tractable conditional distributions for inference about both Lévy measures. We further show that in the Gamma process special case, conjugacy can be achieved for hyper-parameter inference. An explicit characterization of the parameter contour for NVM processes is provided, enabling an identifiable parameterization of the model for effective Markov Chain Monte Carlo algorithms in posterior inference. The method is demonstrated on both synthetic and tick-level (high-frequency) financial datasets.

翻译：摘要：Lévy过程以其能够刻画偏态、重尾和不连续性的复杂动态特性而著称，在跨领域的随机建模中扮演关键角色。然而，无论参数化还是非参数化设定下，多数Lévy过程的推断仍面临重大挑战。本文提出一种新颖的贝叶斯非参数推断框架，用于在线性状态空间模型中推断从属过程与正态方差-均值（NVM）过程的Lévy测度。我们引入了一种灵活的随机测度——独立伽马缩放狄利克雷过程（IGSDP），其中著名的伽马过程是其特例，这使得两种Lévy测度的推断具有可处理的条件分布。进一步证明，在伽马过程特例中，可实现超参数推断的共轭性。针对NVM过程提供了参数轮廓的显式刻画，使模型具有可辨识的参数化形式，从而支持后验推断中有效的马尔可夫链蒙特卡洛算法。该方法在合成数据与逐笔（高频）金融数据上均得到验证。