Conditional independence and graphical models are crucial concepts for sparsity and statistical modeling in higher dimensions. For L\'evy processes, a widely applied class of stochastic processes, these notions have not been studied. By the L\'evy-It\^o decomposition, a multivariate L\'evy process can be decomposed into the sum of a Brownian motion part and an independent jump process. We show that conditional independence statements between the marginal processes can be studied separately for these two parts. While the Brownian part is well-understood, we derive a novel characterization of conditional independence between the sample paths of the jump process in terms of the L\'evy measure. We define L\'evy graphical models as L\'evy processes that satisfy undirected or directed Markov properties. We prove that the graph structure is invariant under changes of the univariate marginal processes. L\'evy graphical models allow the construction of flexible, sparse dependence models for L\'evy processes in large dimensions, which are interpretable thanks to the underlying graph. For trees, we develop statistical methodology to learn the underlying structure from low- or high-frequency observations of the L\'evy process and show consistent graph recovery. We apply our method to model stock returns from U.S. companies to illustrate the advantages of our approach.

翻译：条件独立性与图模型是高维稀疏性与统计建模中的核心概念。对于Lévy过程这类应用广泛的随机过程，这些概念尚未得到系统研究。通过Lévy-Itô分解，多元Lévy过程可分解为布朗运动部分与独立跳跃过程之和。我们证明，边缘过程间的条件独立性可分别针对这两部分进行研究。在布朗运动部分已有完备理论的基础上，我们基于Lévy测度提出了跳跃过程样本路径间条件独立性的全新刻画方法。我们将满足无向或有向马尔可夫性质的Lévy过程定义为Lévy图模型，并证明图结构在单变量边缘过程变化下具有不变性。Lévy图模型能够为高维Lévy过程构建灵活且稀疏的依赖模型，其底层图结构赋予了模型可解释性。针对树状结构，我们开发了从Lévy过程的低频或高频观测数据中学习底层结构的统计方法，并证明了图结构恢复的一致性。通过美国上市公司股票收益率建模的实证研究，我们展示了该方法的优越性。