Lévy graphical models
Conditional independence and graphical models are crucial concepts for sparsity and statistical modeling in higher dimensions. For Lévy processes, a widely applied class of stochastic processes, these notions have not been studied. By the Lévy-Itô decomposition, a multivariate Lévy 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 Lévy measure. We define Lévy graphical models as Lévy processes that satisfy undirected or directed Markov properties. We prove that the graph structure is invariant under changes of the univariate marginal processes. Lévy graphical models allow the construction of flexible, sparse dependence models for Lévy 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 Lévy 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.