Jürgen Schmiegel

Time Change, Volatility, and Turbulence

A concept of volatility modulated Volterra processes is introduced. Apart from some brief discussion of generalities, the paper focusses on the special case of backward moving average processes driven by Brownian motion. In this framework, a review is given of some recent modelling of turbulent velocities and associated questions of time change and universality. A discussion of similarities and differences to the dynamics of financial price processes is included.

Assessing Relative Volatility/Intermittency/Energy Dissipation

We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency even when the data of interest are generated by a non-semimartingale, or a Brownian semistationary process in particular. While this estimation method is motivated by the assessment of relative energy dissipation in empirical data of turbulence, we apply it also to energy price data. Moreover, we develop a probabilistic asymptotic theory for relative power variations of Brownian semistationary processes and Ito semimartingales and discuss how it can be used for inference on relative volatility/intermittency.

Levy-based growth models

In the present paper, we give a condensed review, for the nonspecialist reader, of a new modelling framework for spatio-temporal processes, based on Levy theory. We show the potential of the approach in stochastic geometry and spatial statistics by studying Levy-based growth modelling of planar objects. The growth models considered are spatio-temporal stochastic processes on the circle. As a by product, flexible new models for space-time covariance functions on the circle are provided. An application of the Levy-based growth models to tumour growth is discussed.

Modelling Turbulent Time Series by BSS-Processes

Brownian semi-stationary processes have been proposed as a class of stochastic models for time series of the turbulent velocity field. We show, by detailed comparison, that these processes are able to reproduce the main characteristics of turbulent data. Furthermore, we present an algorithm that allows to estimate the model parameters from second and third order statistics. As an application we synthesise a turbulent time series measured in a helium jet flow.

A causal continuous-time stochastic model for the turbulent energy cascade in a helium jet flow

We discuss continuous cascade models and their potential for modelling the energy dissipation in a turbulent flow. Continuous cascade processes, expressed in terms of stochastic integrals with respect to Lévy bases, are examples of ambit processes. These models are known to reproduce experimentally observed properties of turbulence: the scaling and self-scaling of the correlators of the energy dissipation and of the moments of the coarse-grained energy dissipation. We compare three models: a normal model, a normal inverse Gaussian model, and a stable model. We show that the normal inverse Gaussian model is superior to both, the normal and the stable models, in terms of reproducing the distribution of the energy dissipation; and that the normal inverse Gaussian model is superior to the normal model and competitive with the stable model in terms of reproducing the self-scaling exponents. Furthermore, we show that the presented analysis is parsimonious in the sense that the self-scaling exponents are predicted from the one-point distribution of the energy dissipation, and that the shape of these distributions is independent of the Reynolds number.

A Lévy Based Approach to Random Vector Fields: With a View Towards Turbulence

Abstract Using integration of deterministic, matrix-valued functions with respect to vector-valued, volatility modulated Lévy bases, we construct random vector fields on Rn. In the statistically homogeneous case, the vector field is given as a convolution. With applications to turbulence in mind, the convolution kernel is expressed in terms of the energy spectrum. The theory is applied to atmospheric boundary layer turbulence where, in particular, the Shkarofsky correlation family (a generalisation of the Matérn correlation family) is shown to fit the data well. A modification of the Shkarofsky correlation family, which reproduces a more rigorously derived result on the small-scale behaviour of fully developed turbulence, is introduced. Since turbulence possesses structure across a wide range of length scales, simulation is non-trivial. Using a smooth partition of unity, a simple algorithm is derived to decompose the simulation problem into computationally tractable subproblems. Applications within the wind energy industry are suggested.

Determining source cumulants in femtoscopy with Gram-Charlier and Edgeworth series

Lowest-order cumulants provide important information on the shape of the emission source in femtoscopy. For the simple case of noninteracting identical particles, we show how the fourth-order source cumulant can be determined from measured cumulants in momentum space. The textbook Gram–Charlier series is found to be highly inaccurate, while the related Edgeworth series provides increasingly accurate estimates. Ordering of terms compatible with the Central Limit Theorem appears to play a crucial role even for non-Gaussian distributions.

Some Recent Developments in Ambit Stochastics

Some of the recent developments in the rapidly expanding field of Ambit Stochastics are here reviewed. After a brief recall of the framework of Ambit Stochastics, two topics are considered: (i) Methods of modelling and inference for volatility/intermittency processes and fields; (ii) Universal laws in turbulence and finance in relation to temporal processes. This review complements two other recent expositions.

Edgeworth versus Gram-Charlier series: x-Cumulant and probability density tests

Edgeworth series are often considered the same as Gram-Charlier series in systematic expansions of nongaussian probability distributions. Testing direct approximations of the probability itself as well as of cumulants in coordinate space as functions of measured cumulants in momentum space, we show how the former far outperforms the latter.

Incremental Similarity and Turbulence

This paper discusses the mathematical representation of an empirically observed phenomenon, referred to as Incremental Similarity. We discuss this feature from the viewpoint of stochastic processes and present a variety of non-trivial examples, including those that are of relevance for turbulence modelling.