## Fachbereich Mathematik

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This thesis deals with the following question. Given a moduli space of coherent sheaves on a projective variety with a fixed Hilbert polynomial, to find a natural construction that replaces the subvariety of the sheaves that are not locally free on their support (we call such sheaves singular) by some variety consisting of sheaves that are locally free on their support. We consider this problem on the example of the coherent sheaves on \(\mathbb P_2\) with Hilbert polynomial 3m+1.
Given a singular coherent sheaf \(\mathcal F\) with singular curve C as its support we replace \(\mathcal F\) by locally free sheaves \(\mathcal E\) supported on a reducible curve \(C_0\cup C_1\), where \(C_0\) is a partial normalization of C and \(C_1\) is an extra curve bearing the degree of \(\mathcal E\). These bundles resemble the bundles considered by Nagaraj and Seshadri. Many properties of the singular 3m+1 sheaves are inherited by the new sheaves we introduce in this thesis (we call them R-bundles). We consider R-bundles as natural replacements of the singular sheaves. R-bundles refine the information about 3m+1 sheaves on \(\mathbb P_2\). Namely, for every isomorphism class of singular 3m+1 sheaves there are \(\mathbb P_1\) many equivalence classes of R-bundles. There is a variety \(\tilde M\) of dimension 10 that may be considered as the space of all the isomorphism classes of the non-singular 3m+1 sheaves on \(\mathbb P_2\) together with all the equivalence classes of all R-bundles. This variety is obtained by blowing up the moduli space of 3m+1 sheaves on \(\mathbb P_2\) along the subvariety of singular sheaves. We modify the definition of a 3m+1 family and obtain a notion of a new family over an arbitrary variety S. In particular 3m+1 families of the non-singular sheaves on \(\mathbb P_2\) are families in this sense. New families over one point are either non-singular 3m+1 sheaves or R-bundles. For every variety S we introduce an equivalence relation on the set of all new families over S. The notion of equivalence for families over one point coincides with isomorphism for non-singular 3m+1 sheaves and with equivalence for R-bundles. We obtain a moduli functor \(\tilde{\mathcal M}:(Sch) \rightarrow (Sets)\) that assigns to every variety S the set of the equivalence classes of the new families over S. There is a natural transformation of functors \(\tilde{\mathcal M}\rightarrow \mathcal M\) that establishes a relation between \(\tilde{\mathcal M}\) and the moduli functor \(\mathcal M\) of the 3m+1 moduli problem on \(\mathbb P_2\). There is also a natural transformation \(\tilde{\mathcal M} \rightarrow Hom(\__ ,\tilde M)\), inducing a bijection \(\tilde{\mathcal M}(pt)\cong \tilde M\), which means that \(\tilde M\) is a coarse moduli space of the moduli problem \(\tilde{\mathcal M}\).

Diese Doktorarbeit befasst sich mit Volatilitätsarbitrage bei europäischen Kaufoptionen und mit der Modellierung von Collateralized Debt Obligations (CDOs). Zuerst wird anhand einer Idee von Carr gezeigt, dass es stochastische Arbitrage in einem Black-Scholes-ähnlichen Modell geben kann. Danach optimieren wir den Arbitrage- Gewinn mithilfe des Erwartungswert-Varianz-Ansatzes von Markowitz und der Martingaltheorie. Stochastische Arbitrage im stochastischen Volatilitätsmodell von Heston wird auch untersucht. Ferner stellen wir ein Markoff-Modell für CDOs vor. Wir zeigen dann, dass man relativ schnell an die Grenzen dieses Modells stößt: Nach dem Ausfall einer Firma steigen die Ausfallintensitäten der überlebenden Firmen an, und kehren nie wieder zu ihrem Ausgangsniveau zurück. Dieses Verhalten stimmt aber nicht mit Beobachtungen am Markt überein: Nach Turbulenzen auf dem Markt stabilisiert sich der Markt wieder und daher würde man erwarten, dass die Ausfallintensitäten der überlebenden Firmen ebenfalls wieder abflachen. Wir ersetzen daher das Markoff-Modell durch ein Semi-Markoff-Modell, das den Markt viel besser nachbildet.

Das zentrale Thema dieser Arbeit sind vollständig gekoppelte reflektierte Vorwärts-Rückwärts-Stochastische-Differentialgleichungen (FBSDE). Zunächst wird ein Spezialfall, die teilweise gekoppelten FBSDE, betrachtet und deren Verbindung zur Bewertung Amerikanischer Optionen aufgezeigt. Für die Lösung dieser Gleichung wird Monte-Carlo-Simulation benötigt, daher werden verschiedene Varianzreduktionsmaßnahmen erarbeitet und miteinander verglichen. Im Folgenden wird der allgemeinere Fall der vollständig gekoppelten reflektierten FBSDE behandelt. Es wird gezeigt, wie das Problem der Lösung dieser Gleichungen in ein Optimierungsproblem übertragen werden kann und infolgedessen mit numerischen Methoden aus diesem Bereich der Mathematik bearbeitet werden kann. Abschließend folgen Vergleiche der verschiedenen numerischen Ansätze mit bereits existierenden Verfahren.

Continuous stochastic control theory has found many applications in optimal investment. However, it lacks some reality, as it is based on the assumption that interventions are costless, which yields optimal strategies where the controller has to intervene at every time instant. This thesis consists of the examination of two types of more realistic control methods with possible applications. In the first chapter, we study the stochastic impulse control of a diffusion process. We suppose that the controller minimizes expected discounted costs accumulating as running and controlling cost, respectively. Each control action causes costs which are bounded from below by some positive constant. This makes a continuous control impossible as it would lead to an immediate ruin of the controller. We give a rigorous development of the relevant theory, where our guideline is to establish verification and convergence results under minimal assumptions, without focusing on the existence of solutions to the corresponding (quasi-)variational inequalities. If the impulse control problem can be characterized or approximated by (quasi-)variational inequalities, it remains to solve these equations. In Section 1.2, we solve the stochastic impulse control problem for a one-dimensional diffusion process with constant coefficients and convex running costs. Further, in Section 1.3, we solve a particular multi-dimensional example, where the uncontrolled process is given by an at least two-dimensional Brownian motion and the cost functions are rotationally symmetric. By symmetry, this problem can be reduced to a one-dimensional problem. In the last section of the first chapter, we suggest a new impulse control problem, where the controller is in addition allowed to invest his initial capital into a market consisting of a money market account and a risky asset. The costs which arise upon controlling the diffusion process and upon trading in this market have to be paid out of the controller's bond holdings. The aim of the controller is to minimize the running costs, caused by the abstract diffusion process, without getting ruined. The second chapter is based on a paper which is joint work with Holger Kraft and Frank Seifried. We analyze the portfolio decision of an investor trading in a market where the economy switches randomly between two possible states, a normal state where trading takes place continuously, and an illiquidity state where trading is not allowed at all. We allow for jumps in the market prices at the beginning and at the end of a trading interruption. Section 2.1 provides an explicit representation of the investor's portfolio dynamics in the illiquidity state in an abstract market consisting of two assets. In Section 2.2 we specify this market model and assume that the investor maximizes expected utility from terminal wealth. We establish convergence results, if the maximal number of liquidity breakdowns goes to infinity. In the Markovian framework of Section 2.3, we provide the corresponding Hamilton-Jacobi-Bellman equations and prove a verification result. We apply these results to study the portfolio problem for a logarithmic investor and an investor with a power utility function, respectively. Further, we extend this model to an economy with three regimes. For instance, the third state could model an additional financial crisis where trading is still possible, but the excess return is lower and the volatility is higher than in the normal state.

This thesis is devoted to applying symbolic methods to the problems of decoding linear codes and of algebraic cryptanalysis. The paradigm we employ here is as follows. We reformulate the initial problem in terms of systems of polynomial equations over a finite field. The solution(s) of such systems should yield a way to solve the initial problem. Our main tools for handling polynomials and polynomial systems in such a paradigm is the technique of Gröbner bases and normal form reductions. The first part of the thesis is devoted to formulating and solving specific polynomial systems that reduce the problem of decoding linear codes to the problem of polynomial system solving. We analyze the existing methods (mainly for the cyclic codes) and propose an original method for arbitrary linear codes that in some sense generalizes the Newton identities method widely known for cyclic codes. We investigate the structure of the underlying ideals and show how one can solve the decoding problem - both the so-called bounded decoding and more general nearest codeword decoding - by finding reduced Gröbner bases of these ideals. The main feature of the method is that unlike usual methods based on Gröbner bases for "finite field" situations, we do not add the so-called field equations. This tremendously simplifies the underlying ideals, thus making feasible working with quite large parameters of codes. Further we address complexity issues, by giving some insight to the Macaulay matrix of the underlying systems. By making a series of assumptions we are able to provide an upper bound for the complexity coefficient of our method. We address also finding the minimum distance and the weight distribution. We provide solid experimental material and comparisons with some of the existing methods in this area. In the second part we deal with the algebraic cryptanalysis of block iterative ciphers. Namely, we analyze the small-scale variants of the Advanced Encryption Standard (AES), which is a widely used modern block cipher. Here a cryptanalyst composes the polynomial systems which solutions should yield a secret key used by communicating parties in a symmetric cryptosystem. We analyze the systems formulated by researchers for the algebraic cryptanalysis, and identify the problem that conventional systems have many auxiliary variables that are not actually needed for the key recovery. Moreover, having many such auxiliary variables, specific to a given plaintext/ciphertext pair, complicates the use of several pairs which is common in cryptanalysis. We thus provide a new system where the auxiliary variables are eliminated via normal form reductions. The resulting system in key-variables only is then solved. We present experimental evidence that such an approach is quite good for small scaled ciphers. We investigate further our approach and employ the so-called meet-in-the-middle principle to see how far one can go in analyzing just 2-3 rounds of scaled ciphers. Additional "tuning techniques" are discussed together with experimental material. Overall, we believe that the material of this part of the thesis makes a step further in algebraic cryptanalysis of block ciphers.

This dissertation deals with two main subjects. Both are strongly related to boundary problems for the Poisson equation and the Laplace equation, respectively. The oblique boundary problem of potential theory as well as the limit formulae and jump relations of potential theory are investigated. We divide this abstract into two parts and start with the oblique boundary problem. Here we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in my diploma thesis. Moreover we prove regularization results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkin approximation. Finally we show that the results are applicable to problems from Geomathematics. Now we come to the limit formulae. There we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. The convergence in Lebesgue spaces for integrable functions is already treated in literature. The achievement of this dissertation is this convergence for the weak derivatives of higher orders. Also the layer functions are elements of Sobolev spaces and the surface is a two dimensional suitable smooth submanifold in the three dimensional space. We are considering the potential of the single layer, the potential of the double layer and their first order normal derivatives. Main tool in the proof in Sobolev norm is the uniform convergence of the tangential derivatives, which is proved with help of some results taken from literature. Additionally, we need a result about the limit formulae in the Lebesgue spaces, which is also taken from literature, and a reduction result for normal derivatives of harmonic functions. Moreover we prove the convergence in the Hölder spaces. Finally we give an application of the limit formulae and jump relations. We generalize a known density of several function systems from Geomathematics in the Lebesgue spaces of square integrable measureable functions, to density in Sobolev spaces, based on the results proved before. Therefore we have prove the limit formula of the single layer potential in dual spaces of Soboelv spaces, where also the layer function is an element of such a distribution space.

This study deals with the optimal control problems of the glass tube drawing processes where the aim is to control the cross-sectional area (circular) of the tube by using the adjoint variable approach. The process of tube drawing is modeled by four coupled nonlinear partial differential equations. These equations are derived by the axisymmetric Stokes equations and the energy equation by using the approach based on asymptotic expansions with inverse aspect ratio as small parameter. Existence and uniqueness of the solutions of stationary isothermal model is also proved. By defining the cost functional, we formulated the optimal control problem. Then Lagrange functional associated with minimization problem is introduced and the first and the second order optimality conditions are derived. We also proved the existence and uniqueness of the solutions of the stationary isothermal model. We implemented the optimization algorithms based on the steepest descent, nonlinear conjugate gradient, BFGS, and Newton approaches. In the Newton method, CG iterations are introduced to solve the Newton equation. Numerical results are obtained for two different cases. In the first case, the cross-sectional area for the entire time domain is controlled and in the second case, the area at the final time is controlled. We also compared the performance of the optimization algorithms in terms of the solution iterations, functional evaluations and the computation time.

In the context of inverse optimization, inverse versions of maximum flow and minimum cost flow problems have thoroughly been investigated. In these network flow problems there are two important problem parameters: flow capacities of the arcs and costs incurred by sending a unit flow on these arcs. Capacity changes for maximum flow problems and cost changes for minimum cost flow problems have been studied under several distance measures such as rectilinear, Chebyshev, and Hamming distances. This thesis also deals with inverse network flow problems and their counterparts tension problems under the aforementioned distance measures. The major goals are to enrich the inverse optimization theory by introducing new inverse network problems that have not yet been treated in the literature, and to extend the well-known combinatorial results of inverse network flows for more general classes of problems with inherent combinatorial properties such as matroid flows on regular matroids and monotropic programming. To accomplish the first objective, the inverse maximum flow problem under Chebyshev norm is analyzed and the capacity inverse minimum cost flow problem, in which only arc capacities are perturbed, is introduced. In this way, it is attempted to close the gap between the capacity perturbing inverse network problems and the cost perturbing ones. The foremost purpose of studying inverse tension problems on networks is to achieve a well-established generalization of the inverse network problems. Since tensions are duals of network flows, carrying the theoretical results of network flows over to tensions follows quite intuitively. Using this intuitive link between network flows and tensions, a generalization to matroid flows and monotropic programs is built gradually up.

The goal of this work is the development and investigation of an interdisciplinary and in itself closed hydrodynamic approach to the simulation of dilute and dense granular flow. The definition of “granular flow” is a nontrivial task in itself. We say that it is either the flow of grains in a vacuum or in a fluid. A grain is an observable piece of a certain material, for example stone when we mean the flow of sand. Choosing a hydrodynamic view on granular flow, we treat the granular material as a fluid. A hydrodynamic model is developed, that describes the process of flowing granular material. This is done through a system of partial differential equations and algebraic relations. This system is derived by the kinetic theory of granular gases which is characterized by inelastic collisions extended with approaches from soil mechanics. Solutions to the system have to be obtained to understand the process. The equations are so difficult to solve that an analytical solution is out of reach. So approximate solutions must be obtained. Hence the next step is the choice or development of a numerical algorithm to obtain approximate solutions of the model. Common to every problem in numerical simulation, these two steps do not lead to a result without implementation of the algorithm. Hence the author attempts to present this work in the following frame, to participate in and contribute to the three areas Physics, Mathematics and Software implementation and approach the simulation of granular flow in a combined and interdisciplinary way. This work is structured as follows. A continuum model for granular flow which covers the regime of fast dilute flow as well as slow dense flow up to vanishing velocity is presented in the first chapter. This model is strongly nonlinear in the dependence of viscosity and other coefficients on the hydrodynamic variables and it is singular because some coefficients diverge towards the maximum packing fraction of grains. Hence the second difficulty, the challenging task of numerically obtaining approximate solutions for this model is faced in the second chapter. In the third chapter we aim at the validation of both the model and the numerical algorithm through numerical experiments and investigations and show their application to industrial problems. There we focus intensively on the shear flow experiment from the experimental and analytical work of Bocquet et al. which serves well to demonstrate the algorithm, all boundary conditions involved and provides a setting for analytical studies to compare our results. The fourth chapter rounds up the work with the implementation of both the model and the numerical algorithm in a software framework for the solution of complex rheology problems developed as part of this thesis.

This thesis deals with the application of binomial option pricing in a single-asset Black-Scholes market and its extension to multi-dimensional situations. Although the binomial approach is, in principle, an efficient method for lower dimensional valuation problems, there are at least two main problems regarding its application: Firstly, traded options often exhibit discontinuities, so that the Berry- Esséen inequality is in general tight; i.e. conventional tree methods converge no faster than with order 1/sqrt(N). Furthermore, they suffer from an irregular convergence behaviour that impedes the possibility to achieve a higher order of convergence via extrapolation methods. Secondly, in multi-asset markets conventional tree construction methods cannot ensure well-defined transition probabilities for arbitrary correlation structures between the assets. As a major aim of this thesis, we present two approaches to get binomial trees into shape in order to overcome the main problems in applications; the optimal drift model for the valuation of single-asset options and the decoupling approach to multi-dimensional option pricing. The new valuation methods are embedded into a self-contained survey of binomial option pricing, which focuses on the convergence behaviour of binomial trees. The optimal drift model is a new one-dimensional binomial scheme that can lead to convergence of order o(1/N) by exploiting the specific structure of the valuation problem under consideration. As a consequence, it has the potential to outperform benchmark algorithms. The decoupling approach is presented as a universal construction method for multi-dimensional trees. The corresponding trees are well-defined for an arbitrary correlation structure of the underlying assets. In addition, they yield a more regular convergence behaviour. In fact, the sawtooth effect can even vanish completely, so that extrapolation can be applied.