## Fachbereich Mathematik

### Refine

#### Year of publication

#### Document Type

- Doctoral Thesis (237) (remove)

#### Keywords

#### Faculty / Organisational entity

- Fachbereich Mathematik (237)
- Fraunhofer (ITWM) (2)

Diese Arbeit gehört in die algebraische Geometrie und die Darstellungstheorie und stellt eine Beziehung zwischen beiden Gebieten dar. Man beschäftigt sich mit den abgeleiteten Kategorien auf flachen Entartungen projektiver Geraden und elliptischer Kurven. Als Mittel benutzt man die Technik der Matrixprobleme. Das Hauptergebnis dieser Dissertation ist der folgende Satz: SATZ. Sei X ein Zykel projektiver Geraden. Dann gibt es drei Typen unzerlegbarer Objekte in D^-(Coh_X): - Shifts von Wolkenkratzergarben in einem regulären Punkt; - Bänder B(w,m,lambda), - Saiten S(w). Ganz analog beweist man die Zahmheit der abgeleiteten Kategorien vieler assoziativer Algebren.

The focus of this work has been to develop two families of wavelet solvers for the inner displacement boundary-value problem of elastostatics. Our methods are particularly suitable for the deformation analysis corresponding to geoscientifically relevant (regular) boundaries like sphere, ellipsoid or the actual Earth's surface. The first method, a spatial approach to wavelets on a regular (boundary) surface, is established for the classical (inner) displacement problem. Starting from the limit and jump relations of elastostatics we formulate scaling functions and wavelets within the framework of the Cauchy-Navier equation. Based on numerical integration rules a tree algorithm is constructed for fast wavelet computation. This method can be viewed as a first attempt to "short-wavelength modelling", i.e. high resolution of the fine structure of displacement fields. The second technique aims at a suitable wavelet approximation associated to Green's integral representation for the displacement boundary-value problem of elastostatics. The starting points are tensor product kernels defined on Cauchy-Navier vector fields. We come to scaling functions and a spectral approach to wavelets for the boundary-value problems of elastostatics associated to spherical boundaries. Again a tree algorithm which uses a numerical integration rule on bandlimited functions is established to reduce the computational effort. For numerical realization for both methods, multiscale deformation analysis is investigated for the geoscientifically relevant case of a spherical boundary using test examples. Finally, the applicability of our wavelet concepts is shown by considering the deformation analysis of a particular region of the Earth, viz. Nevada, using surface displacements provided by satellite observations. This represents the first step towards practical applications.

The main two problems of continuous-time financial mathematics are option pricing and portfolio optimization. In this thesis, various new aspects of these major topics of financial mathematics will be discussed. In all our considerations we will assume the standard diffusion type setting for securitiy prices which is today well-know under the term "Black-Scholes model". This setting and the basic results of option pricing and portfolio optimization are surveyed in the first chapter. The next three chapters deal with generalizations of the standard portfolio problem, also know as "Merton's problem". Here, we will always use the stochastic control approach as introduced in the seminal papers by Merton (1969, 1971, 1990). One such problem is the very realistic setting of an investor who is faced with fixed monetary streams. More precisely, in addition to maximizing the utility from final wealth via choosing an investment strategy, the investor also has to fulfill certain consumption needs. Also the opposite situation, an additional income stream can now be taken into account in our portfolio optimization problem. We consider various examples and solve them on one hand via classical stochastic control methods and on the other hand by our new separation theorem. This together with some numerical examples forms Chapter 2. Chapter 3 is mainly concerned with the portfolio problem if the investor has different lending and borrowing rates. We give explicit solutions (where possible) and numerical methods to calculate the optimal strategy in the cases of log utility and HARA utility for three different modelling approaches of the dependence of the borrowing rate on the fraction of wealth financed by a credit. The further generalization of the standard Merton problem in Chapter 4 consists in considering simultaneously the possibilities for continuous and discrete consumption. In our general approach there is a possibility for assigning the different consumption times different weights which is a generalization of the usual way of making them comparable via discounting. Chapter 5 deals with the special case of pricing basket options. Here, the main problem is not path-dependence but the multi-dimensionality which makes it impossible to give usuefull analytical representations of the option price. We review the literature and compare six different numerical methods in a systematic way. Thereby we also look at the influence of various parameters such as strike, correlation, forwards or volatilities on the erformance of the different numerical methods. The problem of pricing Asian options on average spot with average strike is the topic of Chapter 6. We here apply the bivariate normal distribution to obtain an approximate option price. This method proves to be very reliable and e±cient for the valuation of different variants of Asian options on average spot with average strike.

The thesis discusses discrete-time dynamic flows over a finite time horizon T. These flows take time, called travel time, to pass an arc of the network. Travel times, as well as other network attributes, such as, costs, arc and node capacities, and supply at the source node, can be constant or time-dependent. Here we review results on discrete-time dynamic flow problems (DTDNFP) with constant attributes and develop new algorithms to solve several DTDNFPs with time-dependent attributes. Several dynamic network flow problems are discussed: maximum dynamic flow, earliest arrival flow, and quickest flow problems. We generalize the hybrid capacity scaling and shortest augmenting path algorithmic of the static network flow problem to consider the time dependency of the network attributes. The result is used to solve the maximum dynamic flow problem with time-dependent travel times and capacities. We also develop a new algorithm to solve earliest arrival flow problems with the same assumptions on the network attributes. The possibility to wait (or park) at a node before departing on outgoing arc is also taken into account. We prove that the complexity of new algorithm is reduced when infinite waiting is considered. We also report the computational analysis of this algorithm. The results are then used to solve quickest flow problems. Additionally, we discuss time-dependent bicriteria shortest path problems. Here we generalize the classical shortest path problems in two ways. We consider two - in general contradicting - objective functions and introduce a time dependency of the cost which is caused by a travel time on each arc. These problems have several interesting practical applications, but have not attained much attention in the literature. Here we develop two new algorithms in which one of them requires weaker assumptions as in previous research on the subject. Numerical tests show the superiority of the new algorithms. We then apply dynamic network flow models and their associated solution algorithms to determine lower bounds of the evacuation time, evacuation routes, and maximum capacities of inhabited areas with respect to safety requirements. As a macroscopic approach, our dynamic network flow models are mainly used to produce good lower bounds for the evacuation time and do not consider any individual behavior during the emergency situation. These bounds can be used to analyze existing buildings or help in the design phase of planning a building.

Extensions of Shallow Water Equations The subject of the thesis of Michael Hilden is the simulation of floods in urban areas. In case of strong rain events, water can flow out of the overloaded sewer system onto the street and damage the connected houses. The dependable simulation of water flow out of a manhole ("manhole") and over a curb ("curb") is crucial for the assessment of the flood risks. The incompressible 3D-Navier-Stokes Equations (3D-NSE) describe the free surface flow of water accurately, but require expensive computations. Therefore, the less CPU-intensive (factor ca.1/100) Shallow Water Equations (SWE) are usually applied in hydrology. They can be derived from 3D-NSE under the assumption of a hydrostatic pressure distribution via depth-integration and are applied successfully in particular to simulations of river flow processes. The SWE-computations of the flow problems "manhole" and "curb" differ to the 3D-NSE results. Thus, SWE need to be extended appropriately to give reliable forecasts for flood risks in urban areas within reduced computational efforts. These extensions are developed based on physical considerations not considered in the classical SWE. In one extension, a vortex layer on the ground is separated from the main flow representing its new bottom. In a further extension, the hydrostatic pressure distribution is corrected by additional terms due to approximations of vertical velocities and their interaction with the flow. These extensions increase the quality of the SWE results for these flow problems up to the quality level of the NSE results within a moderate increase of the CPU efforts.

The dissertation is concerned with the numerical solution of Fokker-Planck equations in high dimensions arising in the study of dynamics of polymeric liquids. Traditional methods based on tensor product structure are not applicable in high dimensions for the number of nodes required to yield a fixed accuracy increases exponentially with the dimension; a phenomenon often referred to as the curse of dimension. Particle methods or finite point set methods are known to break the curse of dimension. The Monte Carlo method (MCM) applied to such problems are 1/sqrt(N) accurate, where N is the cardinality of the point set considered, independent of the dimension. Deterministic version of the Monte Carlo method called the quasi Monte Carlo method (QMC) are quite effective in integration problems and accuracy of the order of 1/N can be achieved, up to a logarithmic factor. However, such a replacement cannot be carried over to particle simulations due to the correlation among the quasi-random points. The method proposed by Lecot (C.Lecot and F.E.Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity, 15 (1999), pp.342-359) is the only known QMC approach, but it not only leads to large particle numbers but also the proven order of convergence is 1/N^(2s) in dimension s. We modify the method presented there, in such a way that the new method works with reasonable particle numbers even in high dimensions and has better order of convergence. Though the provable order of convergence is 1/sqrt(N), the results show less variance and thus the proposed method still slightly outperforms standard MCM.

Matrix Compression Methods for the Numerical Solution of Radiative Transfer in Scattering Media
(2002)

Radiative transfer in scattering media is usually described by the radiative transfer equation, an integro-differential equation which describes the propagation of the radiative intensity along a ray. The high dimensionality of the equation leads to a very large number of unknowns when discretizing the equation. This is the major difficulty in its numerical solution. In case of isotropic scattering and diffuse boundaries, the radiative transfer equation can be reformulated into a system of integral equations of the second kind, where the position is the only independent variable. By employing the so-called momentum equation, we derive an integral equation, which is also valid in case of linear anisotropic scattering. This equation is very similar to the equation for the isotropic case: no additional unknowns are introduced and the integral operators involved have very similar mapping properties. The discretization of an integral operator leads to a full matrix. Therefore, due to the large dimension of the matrix in practical applcation, it is not feasible to assemble and store the entire matrix. The so-called matrix compression methods circumvent the assembly of the matrix. Instead, the matrix-vector multiplications needed by iterative solvers are performed only approximately, thus, reducing, the computational complexity tremendously. The kernels of the integral equation describing the radiative transfer are very similar to the kernels of the integral equations occuring in the boundary element method. Therefore, with only slight modifications, the matrix compression methods, developed for the latter are readily applicable to the former. As apposed to the boundary element method, the integral kernels for radiative transfer in absorbing and scattering media involve an exponential decay term. We examine how this decay influences the efficiency of the matrix compression methods. Further, a comparison with the discrete ordinate method shows that discretizing the integral equation may lead to reductions in CPU time and to an improved accuracy especially in case of small absorption and scattering coefficients or if local sources are present.

Semiparametric estimation of conditional quantiles for time series, with applications in finance
(2003)

The estimation of conditional quantiles has become an increasingly important issue in insurance and financial risk management. The stylized facts of financial time series data has rendered direct applications of extreme value theory methodologies, in the estimation of extreme conditional quantiles, inappropriate. On the other hand, quantile regression based procedures work well in nonextreme parts of a given data but breaks down in extreme probability levels. In order to solve this problem, we combine nonparametric regressions for time series and extreme value theory approaches in the estimation of extreme conditional quantiles for financial time series. To do so, a class of time series models that is similar to nonparametric AR-(G)ARCH models but which does not depend on distributional and moments assumptions, is introduced. We discuss estimation procedures for the nonextreme levels using the models and consider the estimates obtained by inverting conditional distribution estimators and by direct estimation using Koenker-Basset (1978) version for kernels. Under some regularity conditions, the asymptotic normality and uniform convergence, with rates, of the conditional quantile estimator for strong mixing time series, are established. We study the estimation of scale function in the introduced models using similar procedures and show that under some regularity conditions, the scale estimate is weakly consistent and asymptotically normal. The application of introduced models in the estimation of extreme conditional quantiles is achieved by augmenting them with methods in extreme value theory. It is shown that the overal extreme conditional quantiles estimator is consistent. A Monte Carlo study is carried out to illustrate the good performance of the estimates and real data are used to demonstrate the estimation of Value-at-Risk and conditional expected shortfall in financial risk management and their multiperiod predictions discussed.

Different aspects of geomagnetic field modelling from satellite data are examined in the framework of modern multiscale approximation. The thesis is mostly concerned with wavelet techniques, i.e. multiscale methods based on certain classes of kernel functions which are able to realize a multiscale analysis of the funtion (data) space under consideration. It is thus possible to break up complicated functions like the geomagnetic field, electric current densities or geopotentials into different pieces and study these pieces separately. Based on a general approach to scalar and vectorial multiscale methods, topics include multiscale denoising, crustal field approximation and downward continuation, wavelet-parametrizations of the magnetic field in Mie-representation as well as multiscale-methods for the analysis of time-dependent spherical vector fields. For each subject the necessary theoretical framework is established and numerical applications examine and illustrate the practical aspects.

The immiscible lattice BGK method for solving the two-phase incompressible Navier-Stokes equations is analysed in great detail. Equivalent moment analysis and local differential geometry are applied to examine how interface motion is determined and how surface tension effects can be included such that consistency to the two-phase incompressible Navier-Stokes equations can be expected. The results obtained from theoretical analysis are verified by numerical experiments. Since the intrinsic interface tracking scheme of immiscible lattice BGK is found to produce unsatisfactory results in two-dimensional simulations several approaches to improving it are discussed but all of them turn out to yield no substantial improvement. Furthermore, the intrinsic interface tracking scheme of immiscible lattice BGK is found to be closely connected to the well-known conservative volume tracking method. This result suggests to couple the conservative volume tracking method for determining interface motion with the Navier-Stokes solver of immiscible lattice BGK. Applied to simple flow fields, this coupled method yields much better results than plain immiscible lattice BGK.

One crucial assumption of continuous financial mathematics is that the portfolio can be rebalanced continuously and that there are no transaction costs. In reality, this of course does not work. On the one hand, continuous rebalancing is impossible, on the other hand, each transaction causes costs which have to be subtracted from the wealth. Therefore, we focus on trading strategies which are based on discrete rebalancing - in random or equidistant times - and where transaction costs are considered. These strategies are considered for various utility functions and are compared with the optimal ones of continuous trading.

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