Statistics of Lévy-driven Models

在连续时间中定义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态，并确保高度的数学可处理性，Lévy过程-一类随机过程，包括例如布朗运动，泊松过程和α-稳定Lévy运动-被用作随机驱动力。在这个项目中，我们考虑（多变量）移动平均类型的模型，特别是非常重要的一类连续时间自回归移动平均（CARMA）过程。其目的是大大提高他们的统计和概率属性的理解，并制定一个简洁的理论，统计推断假设的过程中观察到的时间只有在100多个点。因此，几个估计平稳Lévy驱动的移动平均和CARMA过程的定义和分析。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，我们研究的协整的定义在多元Lévy驱动的CARMA框架和相关的统计推断。为了提高我们的估计的适用性，我们研究自助方法。

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.