Statistics of Lévy-driven Models

• Applicants: Professorin Dr. Vicky Fasen-Hartmann; Professor Dr. Alexander Lindner; Professor Dr. Robert Stelzer
• Subject Area: Mathematics
• Term: from 2011 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 190212351

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.

• DFG Programme: Research Grants