## Fachbereich Mathematik

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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.

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.

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.

This thesis is devoted to two main topics (accordingly, there are two chapters): In the first chapter, we establish a tropical intersection theory with analogue notions and tools as its algebro-geometric counterpart. This includes tropical cycles, rational functions, intersection products of Cartier divisors and cycles, morphisms, their functors and the projection formula, rational equivalence. The most important features of this theory are the following: - It unifies and simplifies many of the existing results of tropical enumerative geometry, which often contained involved ad-hoc computations. - It is indispensable to formulate and solve further tropical enumerative problems. - It shows deep relations to the intersection theory of toric varieties and connected fields. - The relationship between tropical and classical Gromov-Witten invariants found by Mikhalkin is made plausible from inside tropical geometry. - It is interesting on its own as a subfield of convex geometry. In the second chapter, we study tropical gravitational descendants (i.e. Gromov-Witten invariants with incidence and "Psi-class" factors) and show that many concepts of the classical Gromov-Witten theory such as the famous WDVV equations can be carried over to the tropical world. We use this to extend Mikhalkin's results to a certain class of gravitational descendants, i.e. we show that many of the classical gravitational descendants of P^2 and P^1 x P^1 can be computed by counting tropical curves satisfying certain incidence conditions and with prescribed valences of their vertices. Moreover, the presented theory is not restricted to plane curves and therefore provides an important tool to derive similar results in higher dimensions. A more detailed chapter synopsis can be found at the beginning of each individual chapter.

This dissertation is intended to give a systematic treatment of hypersurface singularities in arbitrary characteristic which provides the necessary tools, theoretically and computationally, for the purpose of classification. This thesis consists of five chapters: In chapter 1, we introduce the background on isolated hypersurface singularities needed for our work. In chapter 2, we formalize the notions of piecewise-homogeneous grading and we discuss thoroughly non-degeneracy in arbitrary characteristic. Chapter 3 is devoted to determinacy and normal forms of isolated hypersurface singularities. In the first part, we give finite determinacy theorems in arbitrary characteristic with respect to right respectively contact equivalence. Furthermore, we show that "isolated" and finite determinacy properties are equivalent. In the second part, we formalize Arnol'd's key ideas for the computation of normal forms an define the conditions (AA) and (AAC). The last part of Chapter 3 is devoted to the study of normal forms in the general setting of hypersurface singularities imposing neither condition (A) nor Newton-Nondegeneracy. In Chapter 4, we present algorithms which we implement in Singular for the purpose of explicit computation of regular bases and normal forms. In chapter 5, we transfer some classical results on invariants over the field C of complex numbers to algebraically closed fields of characteristic zero known as Lefschetz principle.

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.

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.

Limit theorems constitute a classical and important field in probability theory. In several applications, in particular in demographic or medical contexts, killed Markov processes suggest themselves as models for populations undergoing culling by mortality or other processes. In these situations mathematical research features a general interest in the observable distribution of survivors, which is known as Yaglom limit or quasi-stationary distribution. Previous work often focuses on discrete state spaces, commonly birth-death processes (or with some more flexible localization of the transitions), with killing only on the boundary. The central concerns of this thesis are to describe, for a given class of one dimensional diffusion processes, the quasistationary distributions (if any), and to describe the convergence (or not) of the process conditioned on survival to one of these quasistationary distributions. Rather general diffusion processes on the half-line are considered, where 0 is allowed to be regular or an exit boundary. Very similar techniques are applied in this work in order to derive results on the large time behavior of an exotic measure valued process, which is closely related to so-called point interactions, which have been widely studied in the mathematical physics literature.