# Anita Behme: On stationary solutions of the SDE

# Viktor Benes: Spatio-temporal modelling of point processes

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.

# Vincenzo Ferrazzano: Data manipulation and realized variations test

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.

# Holger Fink: A fractional credit model with long range dependent hazard rate

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.

# Linda Vadgård Hansen: Lévy particles

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.

# Emil Hedevang: Modelling atmospheric boundary layer turbulence using Ambit processes

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.

# Markus Schicks: Finite variation of fractional Lévy processes

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.

# Heikki Tikanmäki: Fractional Lévy processes by integral transformations

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.

# Linda Vos: Levy copula on random fields

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.