Statistics of Lévy-driven Models

定义在连续时间上的时间序列模型对于金融、保险、物理、信号处理和控制等许多应用领域的随机建模非常重要。为了获得真实的边缘分布和动力学，并确保高度的数学可操作性，L过程--一类随机过程，包括例如布朗运动、泊松过程和α稳定的L运动--被用作随机驱动力。在这个项目中，我们考虑了移动平均类型的(多变量)模型，特别是一类非常重要的连续时间自回归移动平均(CARMA)过程。其目的是大大提高对其统计和概率性质的理解，并发展一种简明的统计推断理论，假设这些过程只在有限的多个时间点上被观察到。因此，需要定义和分析平稳的L驱动的移动平均和CARMA过程的几个估计量。由于多变量数据集往往不是平稳的，但某些线性组合是平稳的，我们研究了多元L驱动的CARMA框架中协整的定义和相关的统计推断。为了提高估计量的适用性，我们研究了Bootstrap方法。

Time series models defined in continuous time are important for stochastic modelling in many areas of applications like e.g. finance, insurance, physics, signal processing and control. To obtain realistic marginal distributions and dynamics and to ensure a high mathematical tractability Lévy processes – a class of stochastic processes including, for instance, Brownian motion, Poisson processes and α-stable Lévy motions – are used as the random driving force. In this project we consider (multivariate) models which are of moving average type and especially the very important class of continuous-time autoregressive moving average (CARMA) processes. The aim is to considerably advance the understanding of their statistical and probabilistic properties and to develop a concise theory of statistical inference assuming that the processes are observed only at finitely many points in time. Thus, several estimators for stationary Lévy-driven moving average and CARMA processes are to be defined and analysed. Since often multivariate data sets are not stationary, but certain linear combinations are stationary, we investigate the definition of co-integration in a multivariate Lévy-driven CARMA framework and the related statistical inference. To improve the applicability of our estimators we study bootstrap methods.