## Fachbereich Mathematik

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- Zinsoptimiertes Schuldenmanagement (2014)
- Das zinsoptimierte Schuldenmanagement hat zum Ziel, eine möglichst effiziente Abwägung zwischen den erwarteten Finanzierungskosten einerseits und den Risiken für den Staatshaushalt andererseits zu finden. Um sich diesem Spannungsfeld zu nähern, schlagen wir erstmals die Brücke zwischen den Problemstellungen des Schuldenmanagements und den Methoden der zeitkontinuierlichen, dynamischen Portfoliooptimierung. Das Schlüsselelement ist dabei eine neue Metrik zur Messung der Finanzierungskosten, die Perpetualkosten. Diese spiegeln die durchschnittlichen zukünftigen Finanzierungskosten wider und beinhalten sowohl die bereits bekannten Zinszahlungen als auch die noch unbekannten Kosten für notwendige Anschlussfinanzierungen. Daher repräsentiert die Volatilität der Perpetualkosten auch das Risiko einer bestimmten Strategie; je langfristiger eine Finanzierung ist, desto kleiner ist die Schwankungsbreite der Perpetualkosten. Die Perpetualkosten ergeben sich als Produkt aus dem Barwert eines Schuldenportfolios und aus der vom Portfolio unabhängigen Perpetualrate. Für die Modellierung des Barwertes greifen wir auf das aus der dynamischen Portfoliooptimierung bekannte Konzept eines selbstfinanzierenden Bondportfolios zurück, das hier auf einem mehrdimensionalen affin-linearen Zinsmodell basiert. Das Wachstum des Schuldenportfolios wird dabei durch die Einbeziehung des Primärüberschusses des Staates gebremst bzw. verhindert, indem wir diesen als externen Zufluss in das selbstfinanzierende Modell aufnehmen. Wegen der Vielfältigkeit möglicher Finanzierungsinstrumente wählen wir nicht deren Wertanteile als Kontrollvariable, sondern kontrollieren die Sensitivitäten des Portfolios gegenüber verschiedenen Zinsbewegungen. Aus optimalen Sensitivitäten können in einem nachgelagerten Schritt dann optimale Wertanteile für verschiedenste Finanzierungsinstrumente abgeleitet werden. Beispielhaft demonstrieren wir dies mittels Rolling-Horizon-Bonds unterschiedlicher Laufzeit. Schließlich lösen wir zwei Optimierungsprobleme mit Methoden der stochastischen Kontrolltheorie. Dabei wird stets der erwartete Nutzen der Perpetualkosten maximiert. Die Nutzenfunktionen sind jeweils an das Schuldenmanagement angepasst und zeichnen sich insbesondere dadurch aus, dass höhere Kosten mit einem niedrigeren Nutzen einhergehen. Im ersten Problem betrachten wir eine Potenznutzenfunktion mit konstanter relativer Risikoaversion, im zweiten wählen wir eine Nutzenfunktion, welche die Einhaltung einer vorgegebenen Schulden- bzw. Kostenobergrenze garantiert.

- Worst-Case Portfolio Optimization: Transaction Costs and Bubbles (2015)
- In this thesis we extend the worst-case modeling approach as first introduced by Hua and Wilmott (1997) (option pricing in discrete time) and Korn and Wilmott (2002) (portfolio optimization in continuous time) in various directions. In the continuous-time worst-case portfolio optimization model (as first introduced by Korn and Wilmott (2002)), the financial market is assumed to be under the threat of a crash in the sense that the stock price may crash by an unknown fraction at an unknown time. It is assumed that only an upper bound on the size of the crash is known and that the investor prepares for the worst-possible crash scenario. That is, the investor aims to find the strategy maximizing her objective function in the worst-case crash scenario. In the first part of this thesis, we consider the model of Korn and Wilmott (2002) in the presence of proportional transaction costs. First, we treat the problem without crashes and show that the value function is the unique viscosity solution of a dynamic programming equation (DPE) and then construct the optimal strategies. We then consider the problem in the presence of crash threats, derive the corresponding DPE and characterize the value function as the unique viscosity solution of this DPE. In the last part, we consider the worst-case problem with a random number of crashes by proposing a regime switching model in which each state corresponds to a different crash regime. We interpret each of the crash-threatened regimes of the market as states in which a financial bubble has formed which may lead to a crash. In this model, we prove that the value function is a classical solution of a system of DPEs and derive the optimal strategies.

- Wavelet Modelling of Ionospheric Currents and Induced Magnetic Fields From Satellite Data (2003)
- The thesis is concerned with the modelling of ionospheric current systems and induced magnetic fields in a multiscale framework. Scaling functions and wavelets are used to realize a multiscale analysis of the function spaces under consideration and to establish a multiscale regularization procedure for the inversion of the considered operator equation. First of all a general multiscale concept for vectorial operator equations between two separable Hilbert spaces is developed in terms of vector kernel functions. The equivalence to the canonical tensorial ansatz is proven and the theory is transferred to the case of multiscale regularization of vectorial inverse problems. As a first application, a special multiresolution analysis of the space of square-integrable vector fields on the sphere, e.g. the Earth’s magnetic field measured on a spherical satellite’s orbit, is presented. By this, a multiscale separation of spherical vector-valued functions with respect to their sources can be established. The vector field is split up into a part induced by sources inside the sphere, a part which is due to sources outside the sphere and a part which is generated by sources on the sphere, i.e. currents crossing the sphere. The multiscale technqiue is tested on a magnetic field data set of the satellite CHAMP and it is shown that crustal field determination can be improved by previously applying our method. In order to reconstruct ionspheric current systems from magnetic field data, an inversion of the Biot-Savart’s law in terms of multiscale regularization is defined. The corresponding operator is formulated and the singular values are calculated. Based on the konwledge of the singular system a regularzation technique in terms of certain product kernels and correponding convolutions can be formed. The method is tested on different simulations and on real magnetic field data of the satellite CHAMP and the proposed satellite mission SWARM.

- Volatilitätsarbitrage und ein Markoff-Modell für CDOs (2009)
- 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.

- Vector Field Approximation on Regular Surfaces in Terms of Outer Harmonic Representations (2007)
- The present work deals with the (global and local) modeling of the windfield on the real topography of Rheinland-Pfalz. Thereby the focus is on the construction of a vectorial windfield from low, irregularly distributed data given on a topographical surface. The developed spline procedure works by means of vectorial (homogeneous, harmonic) polynomials (outer harmonics) which control the oscillation behaviour of the spline interpoland. In the process the characteristic of the spline curvature which defines the energy norm is assumed to be on a sphere inside the Earth interior and not on the Earth’s surface. The numerical advantage of this method arises from the maximum-minimum principle for harmonic functions.

- Vector bundles on degenerations of elliptic curves of types II, III and IV (2007)
- In this thesis we classify simple coherent sheaves on Kodaira fibers of types II, III and IV (cuspidal and tacnode cubic curves and a plane configuration of three concurrent lines). Indecomposable vector bundles on smooth elliptic curves were classified in 1957 by Atiyah. In works of Burban, Drozd and Greuel it was shown that the categories of vector bundles and coherent sheaves on cycles of projective lines are tame. It turns out, that all other degenerations of elliptic curves are vector-bundle-wild. Nevertheless, we prove that the category of coherent sheaves of an arbitrary reduced plane cubic curve, (including the mentioned Kodaira fibers) is brick-tame. The main technical tool of our approach is the representation theory of bocses. Although, this technique was mainly used for purely theoretical purposes, we illustrate its computational potential for investigating tame behavior in wild categories. In particular, it allows to prove that a simple vector bundle on a reduced cubic curve is determined by its rank, multidegree and determinant, generalizing Atiyah's classification. Our approach leads to an interesting class of bocses, which can be wild but are brick-tame.

- Variance Reduction Procedures for Market Risk Estimation (2014)
- Monte Carlo simulation is one of the commonly used methods for risk estimation on financial markets, especially for option portfolios, where any analytical approximation is usually too inaccurate. However, the usually high computational effort for complex portfolios with a large number of underlying assets motivates the application of variance reduction procedures. Variance reduction for estimating the probability of high portfolio losses has been extensively studied by Glasserman et al. A great variance reduction is achieved by applying an exponential twisting importance sampling algorithm together with stratification. The popular and much faster Delta-Gamma approximation replaces the portfolio loss function in order to guide the choice of the importance sampling density and it plays the role of the stratification variable. The main disadvantage of the proposed algorithm is that it is derived only in the case of Gaussian and some heavy-tailed changes in risk factors. Hence, our main goal is to keep the main advantage of the Monte Carlo simulation, namely its ability to perform a simulation under alternative assumptions on the distribution of the changes in risk factors, also in the variance reduction algorithms. Step by step, we construct new variance reduction techniques for estimating the probability of high portfolio losses. They are based on the idea of the Cross-Entropy importance sampling procedure. More precisely, the importance sampling density is chosen as the closest one to the optimal importance sampling density (zero variance estimator) out of some parametric family of densities with respect to Kullback - Leibler cross-entropy. Our algorithms are based on the special choices of the parametric family and can now use any approximation of the portfolio loss function. A special stratification is developed, so that any approximation of the portfolio loss function under any assumption of the distribution of the risk factors can be used. The constructed algorithms can easily be applied for any distribution of risk factors, no matter if light- or heavy-tailed. The numerical study exhibits a greater variance reduction than of the algorithm from Glasserman et al. The use of a better approximation may improve the performance of our algorithms significantly, as it is shown in the numerical study. The literature on the estimation of the popular market risk measures, namely VaR and CVaR, often refers to the algorithms for estimating the probability of high portfolio losses, describing the corresponding transition process only briefly. Hence, we give a consecutive discussion of this problem. Results necessary to construct confidence intervals for both measures under the mentioned variance reduction procedures are also given.

- Valuation of Credit Derivatives (2011)
- In this work two main approaches for the evaluation of credit derivatives are analyzed: the copula based approach and the Markov Chain based approach. This work gives the opportunity to use the advantages and avoid disadvantages of both approaches. For example, modeling of contagion effects, i.e. modeling dependencies between counterparty defaults, is complicated under the copula approach. One remedy is to use Markov Chain, where it can be done directly. The work consists of five chapters. The first chapter of this work extends the model for the pricing of CDS contracts presented in the paper by Kraft and Steffensen (2007). In the widely used models for CDS pricing it is assumed that only borrower can default. In our model we assume that each of the counterparties involved in the contract may default. Calculated contract prices are compared with those calculated under usual assumptions. All results are summarized in the form of numerical examples and plots. In the second chapter the copula and its main properties are described. The methods of constructing copulas as well as most common copulas families and its properties are introduced. In the third chapter the method of constructing a copula for the existing Markov Chain is introduced. The cases with two and three counterparties are considered. Necessary relations between the transition intensities are derived to directly find some copula functions. The formulae for default dependencies like Spearman's rho and Kendall's tau for defined copulas are derived. Several numerical examples are presented in which the copulas are built for given Markov Chains. The fourth chapter deals with the approximation of copulas if for a given Markov Chain a copula cannot be provided explicitly. The fifth chapter concludes this thesis.

- Utility-based proof for the existence of strictly consistent price processes under proportional transaction costs (2012)
- This thesis deals with the relationship between no-arbitrage and (strictly) consistent price processes for a financial market with proportional transaction costs in a discrete time model. The exact mathematical statement behind this relationship is formulated in the so-called Fundamental Theorem of Asset Pricing (FTAP). Among the many proofs of the FTAP without transaction costs there is also an economic intuitive utility-based approach. It relies on the economic intuitive fact that the investor can maximize his expected utility from terminal wealth. This approach is rather constructive since the equivalent martingale measure is then given by the marginal utility evaluated at the optimal terminal payoff. However, in the presence of proportional transaction costs such a utility-based approach for the existence of consistent price processes is missing in the literature. So far, rather deep methods from functional analysis or from the theory of random sets have been used to show the FTAP under proportional transaction costs. For the sake of existence of a utility-maximizing payoff we first concentrate on a generic single-period model with only one risky asset. The marignal utility evaluated at the optimal terminal payoff yields the first component of a consistent price process. The second component is given by the bid-ask prices depending on the investors optimal action. Even more is true: nearby this consistent price process there are many strictly consistent price processes. Their exact structure allows us to apply this utility-maximizing argument in a multi-period model. In a backwards induction we adapt the given bid-ask prices in such a way so that the strictly consistent price processes found from maximizing utility can be extended to terminal time. In addition possible arbitrage opportunities of the 2nd kind vanish which can present for the original bid-ask process. The notion of arbitrage opportunities of the 2nd kind has been so far investigated only in models with strict costs in every state. In our model transaction costs need not be present in every state. For a model with finitely many risky assets a similar idea is applicable. However, in the single-period case we need to develop new methods compared to the single-period case with only one risky asset. There are mainly two reasons for that. Firstly, it is not at all obvious how to get a consistent price process from the utility-maximizing payoff, since the consistent price process has to be found for all assets simultaneously. Secondly, we need to show directly that the so-called vector space property for null payoffs implies the robust no-arbitrage condition. Once this step is accomplished we can à priori use prices with a smaller spread than the original ones so that the consistent price process found from the utility-maximizing payoff is strictly consistent for the original prices. To make the results applicable for the multi-period case we assume that the prices are given by compact and convex random sets. Then the multi-period case is similar to the case with only one risky asset but more demanding with regard to technical questions.

- Upscaling Approaches for Nonlinear Processes in Lithium-Ion Batteries (2015)
- Lithium-ion batteries are broadly used nowadays in all kinds of portable electronics, such as laptops, cell phones, tablets, e-book readers, digital cameras, etc. They are preferred to other types of rechargeable batteries due to their superior characteristics, such as light weight and high energy density, no memory effect, and a big number of charge/discharge cycles. The high demand and applicability of Li-ion batteries naturally give rise to the unceasing necessity of developing better batteries in terms of performance and lifetime. The aim of the mathematical modelling of Li-ion batteries is to help engineers test different battery configurations and electrode materials faster and cheaper. Lithium-ion batteries are multiscale systems. A typical Li-ion battery consists of multiple connected electrochemical battery cells. Each cell has two electrodes - anode and cathode, as well as a separator between them that prevents a short circuit. Both electrodes have porous structure composed of two phases - solid and electrolyte. We call macroscale the lengthscale of the whole electrode and microscale - the lengthscale at which we can distinguish the complex porous structure of the electrodes. We start from a Li-ion battery model derived on the microscale. The model is based on nonlinear diffusion type of equations for the transport of Lithium ions and charges in the electrolyte and in the active material. Electrochemical reactions on the solid-electrolyte interface couple the two phases. The interface kinetics is modelled by the highly nonlinear Butler-Volmer interface conditions. Direct numerical simulations with standard methods, such as the Finite Element Method or Finite Volume Method, lead to ill-conditioned problems with a huge number of degrees of freedom which are difficult to solve. Therefore, the aim of this work is to derive upscaled models on the lengthscale of the whole electrode so that we do not have to resolve all the small-scale features of the porous microstructure thus reducing the computational time and cost. We do this by applying two different upscaling techniques - the Asymptotic Homogenization Method and the Multiscale Finite Element Method (MsFEM). We consider the electrolyte and the solid as two self-complementary perforated domains and we exploit this idea with both upscaling methods. The first method is restricted only to periodic media and periodically oscillating solutions while the second method can be applied to randomly oscillating solutions and is based on the Finite Element Method framework. We apply the Asymptotic Homogenization Method to derive a coupled macro-micro upscaled model under the assumption of periodic electrode microstructure. A crucial step in the homogenization procedure is the upscaling of the Butler-Volmer interface conditions. We rigorously determine the asymptotic order of the interface exchange current densities and we perform a comprehensive numerical study in order to validate the derived homogenized Li-ion battery model. In order to upscale the microscale battery problem in the case of random electrode microstructure we apply the MsFEM, extended to problems in perforated domains with Neumann boundary conditions on the holes. We conduct a detailed numerical investigation of the proposed algorithm and we show numerical convergence of the method that we design. We also apply the developed technique to a simplified two-dimensional Li-ion battery problem and we show numerical convergence of the solution obtained with the MsFEM to the reference microscale one.