Survey Talks

Ambit Processes, Ole E. Barndorff-Nielsen
Multipower Variation, Mark Podolskij
CLT in Malliavin Calculus, JosÚ-Manuel Corcuera
Turbulence, JŘrgen Schmiegel
Ambit Processes in Energy Markets, Fred Espen Benth
Infinite Divisibility, Jan Rosinski

 

╗ Ambit Processes, Ole E. Barndorff-Nielsen, Aarhus University, Denmark

Abstract: The field of ambit processes constitutes a general framework for continuous time modelling of tempo-spatial, primarily stationary, processes. A survey is given of the general structure of ambit processes and of a range of the novel mathematical questions to which they give rise. Furthermore, a variety of applications will be indicated. A number of the issues mentioned will be discussed in detail in some of the other talks of the workshop.

 

╗ Multipower Variation, Mark Podolskij, ETHZ Zurich, Switzerland

Abstract: In  the first part of the lecture we present the mathematical theory behind the realised multipower variation. We concentrate on certain functionals of Ito semimartingales based on high-frequency observations. We prove the laws of large numbers and show the associated (stable) central limit theorems. Our focus will be on understanding a pretty complex structure of our statistics and on presenting a simple intuition ("road map") behind the proofs.

In the second part of the lecture we present various statistical applications of the class of multipower variations and related functionals. This class is very useful for different estimation and testing problems regarding the fine structure of a price process in finance. In particular, such characteristics as quadratic variation, volatility or jumps can be estimated by statistics of multipower variation type.


╗ CLT in Malliavin Calculus, JosÚ-Manuel Corcuera, University of Barcelona, Spain

Abstract: Malliavin Calculus is an infinite-dimensional differential Calculus on the Wiener space. Recently, after the work of D. Nualart and collaborators, this calculus has revealed specially useful to derive central limit theorems for functionals of Gaussian Processes. The idea of this survey talk   is to show this theory and its applications in the study of the asymptotic behaviour of the multipower variation of some stochastic processes.

╗ Turbulence, JŘrgen Schmiegel, Aarhus University, Denmark

Abstract: We discuss a stochastic modelling framework for the dynamics of the turbulent velocity field. The model is given by an ambit process in combination with a stochastic intermittency field for the underlying energy dissipation process. Within our modelling framework we are able to reproduce the main stylized facts of turbulent time series.

╗ Ambit Processes in Energy Markets, Fred Espen Benth, University of Oslo, Norway

Title: Modelling energy markets by ambit processes

Abstract: We discuss the use of different ambit-like processes to model spot and forward prices in energy markets. Stylized features of energy spot prices are seasonality, stationarity or mean-reversion, and spikes.
These can all be accounted for in general classes of models. On the other hand, motivated from the Heath-Jarrow-Morton we discuss various extensions inspired by ambit processes to forward price modelling. We discuss the link between forward and spot prices in this framework. This is a survey talk discussing various modelling issues in energy markets and how ambit processes may provide a natural class of models.

The talk is based on joint work with Ole Barndorff-Nielsen and Almut Veraart.

╗ Infinite Divisible Processes - an Overview, Jan Rosinski, University of Tennessee, USA

Abstract: Infinitely divisible (ID) processes can be defined on any index set, similarly to Gaussian processes. Their most prominent members, LÚvy processes, are at the heart of modern probability theory and stochastic modeling. However, Levy processes constitute only a small subclass of stationary increment ID processes. Stationary increment and stationary ID processes  form large classes of ID models with a variety of  sample path properties and rich dependence and memory structures. A complete and workable characterization of such processes is still out of reach.

In this talk we will give an overview for a systematic treatment of ID processes with stress on stationary and stationary increment processes. We will present characterizations and various representations for special classes of ID processes, including stable, tempered stable, and selfdecomposable processes. Certain representations lead to extensions of results for ID processes to LÚvy chaos.  Tools used in this area range from probability in Banach spaces to infinite ergodic theory.