## Dissertation

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#### Erscheinungsjahr

#### Dokumenttyp

- Dissertation (226) (entfernen)

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#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (226) (entfernen)

In this work we present and estimate an explanatory model with a predefined system of explanatory equations, a so called lag dependent model. We present a locally optimal, on blocked neural network based lag estimator and theorems about consistensy. We define the change points in context of lag dependent model, and present a powerfull algorithm for change point detection in high dimensional high dynamical systems. We present a special kind of bootstrap for approximating the distribution of statistics of interest in dependent processes.

In the present work, we investigated how to correct the questionable normality, linear and quadratic assumptions underlying existing Value-at-Risk methodologies. In order to take also into account the skewness, the heavy tailedness and the stochastic feature of the volatility of the market values of financial instruments, the constant volatility hypothesis widely used by existing Value-at-Risk appproches has also been investigated and corrected and the tails of the financial returns distributions have been handled via Generalized Pareto or Extreme Value Distributions. Artificial Neural Networks have been combined by Extreme Value Theory in order to build consistent and nonparametric Value-at-Risk measures without the need to make any of the questionable assumption specified above. For that, either autoregressive models (AR-GARCH) have been used or the direct characterization of conditional quantiles due to Bassett, Koenker [1978] and Smith [1987]. In order to build consistent and nonparametric Value-at-Risk estimates, we have proved some new results extending White Artificial Neural Network denseness results to unbounded random variables and provide a generalisation of the Bernstein inequality, which is needed to establish the consistency of our new Value-at-Risk estimates. For an accurate estimation of the quantile of the unexpected returns, Generalized Pareto and Extreme Value Distributions have been used. The new Artificial Neural Networks denseness results enable to build consistent, asymptotically normal and nonparametric estimates of conditional means and stochastic volatilities. The denseness results uses the Sobolev metric space L^m (my) for some m >= 1 and some probability measure my and which holds for a certain subclass of square integrable functions. The Fourier transform, the new extension of the Bernstein inequality for unbounded random variables from stationary alpha-mixing processes combined with the new generalization of a result of White and Wooldrige [1990] have been the main tool to establich the extension of White's neural network denseness results. To illustrate the goodness and level of accuracy of the new denseness results, we were able to demonstrate the applicability of the new Value-at-Risk approaches by means of three examples with real financial data mainly from the banking sector traded on the Frankfort Stock Exchange.

This thesis builds a bridge between singularity theory and computer algebra. To an isolated hypersurface singularity one can associate a regular meromorphic connection, the Gauß-Manin connection, containing a lattice, the Brieskorn lattice. The leading terms of the Brieskorn lattice with respect to the weight and V-filtration of the Gauß-Manin connection define the spectral pairs. They correspond to the Hodge numbers of the mixed Hodge structure on the cohomology of the Milnor fibre and belong to the finest known invariants of isolated hypersurface singularities. The differential structure of the Brieskorn lattice can be described by two complex endomorphisms A0 and A1 containing even more information than the spectral pairs. In this thesis, an algorithmic approach to the Brieskorn lattice in the Gauß-Manin connection is presented. It leads to algorithms to compute the complex monodromy, the spectral pairs, and the differential structure of the Brieskorn lattice. These algorithms are implemented in the computer algebra system Singular.

The study of families of curves with prescribed singularities has a long tradition. Its foundations were laid by Plücker, Severi, Segre, and Zariski at the beginning of the 20th century. Leading to interesting results with applications in singularity theory and in the topology of complex algebraic curves and surfaces it has attained the continuous attraction of algebraic geometers since then. Throughout this thesis we examine the varieties V(D,S1,...,Sr) of irreducible reduced curves in a fixed linear system |D| on a smooth projective surface S over the complex numbers having precisely r singular points of types S1,...,Sr. We are mainly interested in the following three questions: 1) Is V(D,S1,...,Sr) non-empty? 2) Is V(D,S1,...,Sr) T-smooth, that is smooth of the expected dimension? 3) Is V(D,S1,...Sr) irreducible? We would like to answer the questions in such a way that we present numerical conditions depending on invariants of the divisor D and of the singularity types S1,...,Sr, which ensure a positive answer. The main conditions which we derive will be of the type inv(S1)+...+inv(Sr) < aD^2+bD.K+c, where inv is some invariant of singularity types, a, b and c are some constants, and K is some fixed divisor. The case that S is the projective plane has been very well studied by many authors, and on other surfaces some results for curves with nodes and cusps have been derived in the past. We, however, consider arbitrary singularity types, and the results which we derive apply to large classes of surfaces, including surfaces in projective three-space, K3-surfaces, products of curves and geometrically ruled surfaces.

Structure and Construction of Instanton Bundles on P3