Ambit processes » Posters

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An idea from ambit processes was used to the Lévy-based Cox point process modelling in space and time. Some aspects of related models are discussed, e.g. the use of stochastic differential equations for either the temporal intensity of the process or for the spatial trajectory in case of a point process on a curve. Markov chain and sequential Monte Carlo methods in parameter estimation of the models are compared and an application in neuroscience is mentioned.

We uncovered that the usual data manipulation performed collecting windspeed
data, can deceive the Barndorf-Nielsen et al. test, based on the ratio between realized quantities. Those operation, like filtering and discretization, are performed virtually on every data regarding physical measures, especially when electronic measurement device are involved.

Motivated by empirical evidence of long range dependence in macroeconomic
variables like interest rates, domestic gross products or supply and demand
rates, we propose a Vasicek model with time dependent coefficient functions
driven by fractional Brownian motion (fBm) to describe the dynamics of the
short rate in a bond market. We calculate the prices of the corresponding
zero-coupon bonds with the help of Wick calculus.

Furthermore we introduce credit risk in our model by a default rate process
positively correlated to the short rate. Its dynamics are described using
again a Vasicek model driven by another fBm. Here we assume that both, short
and default rate, are driven by the same market noise. We solve the
mathematical problem to predict exponentials of two fBm integrals at the
same time and find closed formulas for the prices of defaultable zero-coupon
bonds. Exploiting Fourier methods we derive option prices in our credit
market and compare them to classical Brownian Vasicek models.

A compact star-shaped subset of R^3 can be described using for instance the radial function or the support function (if the set is also convex). We propose modelling random compact star-shaped (convex) sets by means of modelling the radial (support) function as a kernel smoothing of a Lévy basis. We suggest estimation using the method of moments or likelihood estimation. Furthermore, we reflect on which model to choose depending on the type observations available, if both are applicable.

The turbulence in atmospheric boundary layer flow is characterised by several statistics, for example the Fourier power spectrum of the velocity, the two-point correlation of the energy dissipation, and distributions of the velocity increments over various lags, to name a few. Jürgen Schmiegel and Ole E. Barndorff-Nielsen have developed a model for the turbulent velocities using Ambit processes and this model is able to reproduce many of these statistics. On the poster I will describe the model and compare it to measured high-frequency wind data.

Fractional Lévy processes are shown to be semimartingales if and only if its
sample paths are of ÿÿnite variation or diÿÿerentiable. This is also equivalent to an
integrability condition on the Lévy measure of the driving Lévy process without
Brownian part. The proofs are entirely based on the stationary increments property
and extend to well-balanced fractional Lévy processes.

There have been several suggestions for the definition of fractional Lévy processes in the literature. In [1] the author defines fractional Lévy processes by replacing Brownian motion by more general Lévy process in infinitely supported Mandelbrot-Van Ness integral representation of fractional Brownian motion.

In this work, a new definition for fractional Lévy processes is introduced by taking compact interval Molchan-Golosov representation of fractional Brownian motion. The driving Brownian motion is replaced by more general Lévy process analogously to the approach of [1]. It is proved that the fractional Lévy processes by the different integral transformations are not the same in distribution in general. However, a connection between the two fractional Lévy process concepts is presented. The properties of the processes are analysed and compared.

References
[1] Tina Marquardt. Fractional Lévy processes with an application to long memory moving average processes. Bernoulli, 12(6):1099-1126, 2006.

We introduce a bivariate Lévy copula on random fields in Hilbert spaces.  The defined notion is a natural extension of the already existing positive Lévy copula. We prove existence and several properties of the object. Moreover we give a method to simulate two-dimensional random fields according to the dependence described by the copula. The copula can have application in modeling dependencies between several bond markets or the dependence between different forward markets of commodities. Especially in electricity market correlated jumps between several forward markets are observed.

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