The aim of this paper is to develop estimation and inference methods for the drift parameters of multivariate L\'evy-driven continuous-time autoregressive processes of order $p\in\mathbb{N}$. Starting from a continuous-time observation of the process, we develop consistent and asymptotically normal maximum likelihood estimators. We then relax the unrealistic assumption of continuous-time observation by considering natural discretizations based on a combination of Riemann-sum, finite difference, and thresholding approximations. The resulting estimators are also proven to be consistent and asymptotically normal under a general set of conditions, allowing for both finite and infinite jump activity in the driving L\'evy process. When discretizing the estimators, allowing for irregularly spaced observations is of great practical importance. In this respect, CAR($p$) models are not just relevant for "true" continuous-time processes: a CAR($p$) specification provides a natural continuous-time interpolation for modeling irregularly spaced data - even if the observed process is inherently discrete. As a practically relevant application, we consider the setting where the multivariate observation is known to possess a graphical structure. We refer to such a process as GrCAR and discuss the corresponding drift estimators and their properties. The finite sample behavior of all theoretical asymptotic results is empirically assessed by extensive simulation experiments.

翻译：本文旨在为$p\in\mathbb{N}$阶多变量Lévy驱动的连续时间自回归过程的漂移参数发展估计与推断方法。基于过程的连续时间观测，我们构建了相合且渐近正态的最大似然估计量。随后，我们通过考虑基于黎曼和、有限差分与阈值近似相结合的自然离散化方法，放松了连续时间观测这一不切实际的假设。在驱动Lévy过程同时具有有限与无限跳跃活动性的一般条件下，所得到的估计量被证明同样具有相合性与渐近正态性。在离散化估计量时，允许非等间隔观测具有重要的实际意义。在此方面，CAR($p$)模型不仅适用于"真实"连续时间过程：CAR($p$)规范为建模非等间隔数据提供了一种自然的连续时间插值方法——即使观测过程本质上是离散的。作为一个具有实际意义的应用，我们考虑了多变量观测已知具有图结构的情形。我们将此类过程称为GrCAR，并讨论了相应的漂移估计量及其性质。通过广泛的仿真实验，对所有理论渐近结果的有限样本行为进行了实证评估。