Volatility modulated Volterra processes
A concept of volatility modulated Volterra processes is introduced, and the question of inference on the (integrated squared) volatility is discussed, particularly for two types of base models, from finance and turbulence, respectively. Finally, the relations to the concept of realised variation ratio will be indicated.
Gaussian Semimartingales and Moving Averages
Continuous time moving average processes, e.g. the fractional Brownian motion or the (generalized) Ornstein-Uhlenbeck process, have been used repeatedly in finance but recently also as a model for the turbulent velocity field by Barndorff-Nielsen and Schmiegel. For natural reasons it is often important that the process of interest is a semimartingale. However, we are far from a complete understanding of when a moving average process is a semimartingale. We present some answers to the above question in the case where the driving process is Brownian motion (as it is for the fBm and the OU-process). Such processes are in particular Gaussian. Moreover, we provide a decomposition result for general Gaussian semimartingales and apply it to obtain a necessary and sufficient condition on the covariance function for a centered Gaussian process to be a semimartingale.
Modelling the electricity markets
Electricity prices are mean-reverting with frequently occuring spikes. We discuss different modelling issues for electricity prices, and analyse these models with a view towards forwards and options.
Uniqueness of Solutions to the Stochastic Navier-Stokes, the Invariant Measure and Kolmogorov's Theory
The existence and uniqueness of solutions of the Navier-Stokes equation driven with additive noise in three dimensions is proven, in the presence of uni-directional mean flow and some swirl. The existence of a unique invariant measure is established and the properties of this measure are described. The invariant measure is used to prove Kolmogorov's scaling in 3-dimensional turbulence including the celebrated -5/3 power law for the decay of the power spectrum of a turbulent 3-dimensional flow.
Multifractal design of wind fields
A good modelling of turbulent wind fields is required to determine the loads and the design of wind turbines. The standard modeling is based on Gaussian fluctuations only. By referring to multifractal stochastic processes, we present a non-Gaussian generalization. It is able to correctly describe the observed higher-order statistics, including for example the spatial and temporal correlation of mini-gusts.
Nonparametric tests for analyzing the fine structure of price fluctuations
Using a nonparametric threshold estimator for the continuous component of the quadratic variation ("integrated variance"), we design a test for the presence of a continuous component in the price process and a test for establishing whether the jump component has finite or infinite variation, based on observations on a discrete time grid. Using simulations of stochastic models commonly used in finance, we confirm the performance of our tests and compare them with analogous tests constructed using multipower variation estimators of integrated variance. Finally, we apply our tests to investigate the fine structure of the DM/USD exchange rate process and of SPX futures prices. In both cases, our tests reveal the presence of a non-zero Brownian component, combined with a finite variation jump component.
Joint work with Cecilia Mancini
Power variation and Gaussian processes with stationary increments
The purpose of this talk is to study the asymptotic behaviour of the observed power variation for processes that are obtained as path-wise integrals with respect to a Gaussian processes with stationary increments. We will also explore the behaviour of other more general functionals of the process.
Homogeneous and isotropic statistical solutions of the Navier-Stokes equations
Constructions of homogeneous and isotropic statistical solutions of the 3D Navier-Stokes system will be presented. It will be shown how to approximate (in the sense of convergence of characteristic functionals) any isotropic measure on a certain space of vector fields by isotropic measures supported by periodic vector fields and their rotations. Obtained homogeneous and isotropic statistical Navier-Stokes solutions are supported by weak solutions of the Navier-Stokes equations. The restriction of the constructed statistical solution at t = 0 is well defined and coincides with the initial measure. The result is published in [1].
[1] S. Dostoglou, A.V. Fursikov, J.D. Kahl, Homogeneous and Isotropic Statistical Solutions of the Navier-Stokes Equations. Math. Physics Electronic Journal, www.ma.utexas.edu/mpej/, volume 12, paper No. 2, 2006.
Modelling and optimization of wind farms
Based on a simple static wind-flow model, an optimal wind-farm control is proposed, where the wind turbines cooperate with each other to increase the power efficiency of the wind farm. For further improvement, a dynamical wake modelling of the intra-farm wind flows is needed. First steps in this direction are presented.
Continuous Time Random Walks in the Continuum Limit: Simulation of Atmospheric Winds
Continuous Time Random Walks (CTRWs) by Montroll and Weiss originally have been defined on a grid exhibiting individual jumps of finite length at every hopping event. In the long run, they can exhibit anomalous diffusion properties in the ensemble sense. Recently, independently from one another at least three groups developed methods to perform the continuum limit of such processes [1-3]. Then, e.g. continuous sample paths of processes with subdiffusive properties efficiently can be generated.
We give a short overview over the concepts behind the different approaches and present trajectories of continuous sample paths. The properties of these paths are discussed and the subdiffusive character is verified. Finally, the prospects for the application of CTRWs for the simulation of turbulent atmospheric wind velocities are discussed.
[1] M. Magdziarz et al., Physical Review E 75, 015708, 2007.
[2] R. Gorenflo et al., Chaos, Solitons and Fractals 34, 87-103, 2007.
[3] D. Kleinhans et al., Physical Review E 76, 061102, 2007.
The distribution of turbulence driven wind speed extremes - an asymptotic closed form formulation
The statistical distribution of extreme wind speed excursions above a mean level, for a specified (large) recurrence period, is of crucial importance in relation to design of wind sensitive structures. This is particularly true for wind turbine structures.
Assuming the stochastic (wind speed) process to be a Gaussian process, Cartwright and Longuet-Higgens [1] derived an asymptotic expression for the distribution of the largest excursion from the mean level during an arbitrary recurrence period. From its inception, this celebrated expression has been widely used in wind engineering (as well as in off-shore engineering) - often through definition of the peak factor, which equates the mean of the Cartwright/Longuet-Higgens asymptotic distribution. However, investigations of full-scale wind speed time series, recorded in the atmospheric boundary layer, has revealed that the Gaussian assumption is inadequate for wind speed events associated with large excursions from the mean [2], [3], [4]. Such extreme turbulence excursions seem to occur significantly more frequent than predicted according to the Gaussian assumption, which may under-predict the probability of large turbulence excursions by more than one decade. This obviously has unfortunate consequences for the applicability of the Cartwright/Lounguet-Higgens asymptotic extreme distribution in the description of extreme turbulence excursions, especially for long recurrence periods. Another related problem with the Cartwright/Longuet-Higgens expression, associated with description of extreme wind speed events in the atmospheric boundary layer, is, that many investigations of full-scale wind speed gusts (e.g. [5], [6]) have shown, that the observed occurrences of these are excellently described by the Gumbel EV1 distribution, which, on the other hand, differs from the asymptotic Cartwright/Longuet-Higgens distribution.
We present an asymptotic expression for the distribution of the largest excursion from the mean level, during a large but otherwise arbitrary recurrence period, based on a Generalised Hyperbolic type of "mother" distribution that reflects the Exponential-like distribution behaviour of large wind speed excursions. The derived asymptotic distribution is shown to equal a Gumbel EV1 type distribution, and the associated two distribution parameters are expressed as simple functions of basic parameters characterizing stochastic wind speed processes in the atmospheric boundary layer.
[1] D.E. Cartwright and M. S. Longuet-Higgins (1956). The statistical distribution of the maxima of a random function.
[2] M. Nielsen, G.C. Larsen, J. Mann, S. Ott, K.S. Hansen and B.J. Pedersen (2003). Wind Simulation for Extreme and Fatigue Loads. Risø-R-1437(EN).
[3] F. Boettcher, C. Renner, H.-P. Waldl, and J. Peinke (2003). On the statistics of Wind Gusts. Boundary Layer Meteorology, 108, 163-173.
[4] H.A. Panofsky and J.A. Dutton (1984). Atmospheric Turbulence - Models and Methods for Engineering Applications. John Wiley & Sons.
[5] G.C. Larsen, K.S. Hansen and B.J. Pedersen (2002). Constrained simulation of critical wind speed gusts by means of wavelets. 2002 Global Windpower Conference and Exhibition, France.
[6] G.C. Larsen and K.S. Hansen (2001). Statistics of Off Shore Wind Speed Gusts. EWEC'01, Copenhagen, Denmark, 2-6 July.
New insights into turbulence with excursion to finance
We present a more complete analysis of disordered systems with a high degree of complexity like measurement data of fully developed, or financial data. In particular the parameter free estimation of the stochastic equations (Fokker-Plack-Kolmogorov equation or Langevin equation) by means of the Kramers-Moyal coefficients provide access to the joint probability density function of increments for n-scales [1]. In this contribution we report on new findings based on this technique and based on the investigation of many different data sets.
In particular we show:
(1) An improved method to reconstruct from given data the underlying stochastic process in form of a Fokker-Planck equation, which includes intermittency effects, will be shown.
(2) It is shown that a new length scale, lmar, can be defined, which corresponds to a memory effect in the cascade dynamics. We call this length Einstein-Markovian coherence length and present some interesting interpretations. [2]
(3) It is shown that the stochastic process of a cascade will change with the Re-number and that it has some non-universal contributions. [3]
(3) A method is presented which allows to reconstruct time series from the estimated stochastic process evolving in scale (see (1)). The original and reconstructed time series coincide with respect to the unconditional and conditional probability densities. Therefore the method proposed here is able to generate artificial time series with correct n-point statistics. [4]
[1] Ch. Renner, J. Peinke & R. Friedrich: Markov properties of small scale turbulence. J. Fluid Mech., 433, 383 (2001)
[2] St. Lück, Ch. Renner, J. Peinke, and R. Friedrich: The Markov coherence length a new meaning for the Taylor length in turbulence. Phys. Lett., in press.
[3] Ch. Renner, J. Peinke, R. Friedrich, O. Chanal, and B. Chabaud: Universality of Small Scale Turbulence. Phys. Rev. Lett., 89, 124502 (2002).
[4] A.P. Nawroth and J. Peinke: Multiscale reconstruction of time series. Phys. Lett., in press.
Inference for semimartingales in the presence of noise
We present estimates for some characteristics of semimartingales, such as quadratic variation, in the presence of a general noise process. The core of our approach is the power variation applied to some moving average quantity. We present different "weak laws of large numbers" according to whether the underlying semimartingale is continuous or not. Furthermore, under very weak assumptions on the semimartingale, we prove the associated central limit theorems.
All central limit theorems have the rate n^{-1/4} and the limiting variables/processes are mixed normal. The asymptotic results can be transformed into feasible (standard) central limit theorems. This theory can be applied to estimate the integrated volatility, squared jumps or for constructing tests for jumps in the presence of noise.
Stochastic modelling of the turbulent velocity field
We discuss a stochastic differential equation, as a modelling framework for the timewise dynamics of turbulent velocities. The equation is capable of capturing basic stylized facts of the statistics of temporal velocity increments.
In particular, we focus on scaling of structure functions and the evolution of the probability density of velocity increments, characterized by a normal inverse Gaussian shape with heavy tails for small scales and approximately Gaussian tails for large scales.
In addition, we show that the proposed model is in accordance with the experimental verification of Kolmogorov's refined similarity hypotheses.
Measuring downside risk - realised semivariance
No abstract available
Efficient and explicit martingale estimating functions for SDE models
Optimal martingale estimating functions have turned out to provide simple estimators for many SDE models of the diffusion type. These estimating functions are approximations to the score functions, which are rarely explicit known, and have often turned out to provide estimators with a surprisingly high efficiency. This can now be explained: the estimators are, under weak conditions, rate optimal and fully efficient in a high frequency asymptotics that is relevant to data from finance as well as turbulence. Statistical inference based on optimal martingale estimating functions is particularly easy when the estimating functions are explicit. This is the case for the class of Pearson diffusions, where the drift is linear and the squared diffusion coefficient is quadratic. This is a surprisingly versatile class of diffusion models. Also for some non-Markovian models based on Pearson diffusions, e.g. stochastic volatility models, explicit estimating functions are available.
Analysis of Ruin Probability under investment for non Markovian interarrival times
Recent work in the actuarial and financial literature illustrates the different behavior of the probability of ruin when the capital is invested in a risky asset. In this talk I will describe recent work with Corina Constantinescu, currently at RIMS, where we obtain an integro-differential equation for the ruin probability under investments, when the distribution of the interrival times of the claims have a density satisfying a differential equation. This case includes most of the known examples in the literature. Using Tauberian type theorems, estimates on the decay of the ruin probability as the initial capital tends to infinity can be obtained.
Inference for the jump part of quadratic variation of Itô semimartingales
When asset prices are modelled by Itô semimartingales, their quadratic variation consists ofa continuous and a jump component. This paper is about inference on the jump part of the quadratic variation, which we estimate by using the difference of realised variance and realised multipower variation. The main contribution of this paper is that we provide a bivariate asymptotic limit theory for realised variance and realised multipower variation in the presence of jumps. From that result, we can then deduce the asymptotic distribution of the estimator of the jump component of quadratic variation and can make inference on it. Furthermore, we present consistent estimators for the asymptotic variances of the limit distributions which allows us to derive a feasible asymptotic theory. Monte Carlo studies reveal a good finite sample performance of the proposed feasible limit theory, and an empirical study shows the relevance of our result in practice.
A Rate of Convergence for the LANSalpha Regularization of Navier-Stokes Equations
The Lagrangian Averaged Navier-Stokes (LANSalpha) equations with periodic boundary were introduced as a regularization of incompressible Navier-Stokes in connection with numerical models of turbulent flows. The analysis of these equations has subsequently become a focus of mathematical research. By means of a probabilistic multiplicative cascade representation for which there is a common probability space for each $\alpha > 0$, one may obtain existence of unique global (in time) solutions together with convergence of the solutions of the $\lansalpha$ equations to the solutions of the Navier-Stokes equations as $\alpha\downarrow 0$.
Under further moment type conditions for majorizing kernels, a rate of convergence in a mixed $L^1-L^2$ time-space norm is obtained as $\alpha\downarrow 0$ for small initial data in a subclass of such spaces. In this talk we will try to present the big picture for this line of research, including both strengths and limitations.
This is based on joint work with Larry Chen, Sun-Chul Kim, Ronald Guenther, and Enrique Thomann at Oregon State University, partially supported by a grant from the National Science Foundation.
Stochastic multiscale analysis and reconstruction of time series of stochastic cascade processes
We report on a stochastic approach for the analysis and reconstruction of complex systems, based on the theory of Markov processes. With this analysis we achieve a characterization of the scale dependent complexity of stochastic cascade processes (such as data from turbulent systems or finanial markets) by means of a Fokker-Planck or Langevin equation, providing the complete stochastic information of multiscale joint probabilities. As a recent result, a new method for the simulation of time series based on these multiscale joint probabilities is presented. It enables the synthetic generation or continuation of time series, preserving the correct statistics of the original system. Furthermore, by forecasting the probability distribution of future values, also changes in variance and other properties of the time series are predicted.
Brownian motion based versus fractional Brownian motion based models
Brownian motion based models, such as diffusion processes or stochastic differential equations are popular models boths in finance and natural sciences. However, in finance empirical studies have shown that tick-by-tick data does not behave like the theory for Brownian motion based stochastic volatility models suggest. As a way to explain these empirical findings the concept of market microstructure has been introduced, which means that a noise term, i.e. a sequence of random variables, mostly assumed to be iid, is added. A different way to explain this behaviour is to consider fractional Brownian motion based models. We develop a test which allows to test simultaneously for Brownian motion based models and Brownian motion based models with iid noise component against fractional Brownian motion based models. We apply our results to Daimler Chrysler and Infineon tick-by-tick data and obtain that we have to reject both hypothesis of Brownian motion based models and iid market microstructure noise. Furthermore, we apply our test to temperature data on different time scales and obtain similar results as for the financial data.
Stationary Solutions of SPDEs and Infinite Horizon BDSDEs
In this talk, I will study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between $L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.
