An increasing number of nowadays tasks, such as speech recognition, image generation, translation, classification or prediction are performed with the help of machine learning. Especially artificial neural networks (ANNs) provide convincing results for these tasks. The reasons for this success story are the drastic increase of available data sources in our more and more digitalized world as well as the development of remarkable ANN architectures. This development has led to an increasing number of model parameters together with more and more complex models. Unfortunately, this yields a loss in the interpretability of deployed models. However, there exists a natural desire to explain the deployed models, not just by empirical observations but also by analytical calculations. In this thesis, we focus on variational autoencoders (VAEs) and foster the understanding of these models. As the name suggests, VAEs are based on standard autoencoders (AEs) and therefore used to perform dimensionality reduction of data. This is achieved by a bottleneck structure within the hidden layers of the ANN. From a data input the encoder, that is the part up to the bottleneck, produces a low dimensional representation. The decoder, the part from the bottleneck to the output, uses this representation to reconstruct the input. The model is learned by minimizing the error from the reconstruction. In our point of view, the most remarkable property and, hence, also a central topic in this thesis is the auto-pruning property of VAEs. Simply speaking, the auto-pruning is preventing the VAE with thousands of parameters from overfitting. However, such a desirable property comes with the risk for the model of learning nothing at all. In this thesis, we look at VAEs and the auto-pruning from two different angles and our main contributions to research are the following: (i) We find an analytic explanation of the auto-pruning. We do so, by leveraging the framework of generalized linear models (GLMs). As a result, we are able to explain training results of VAEs before conducting the actual training. (ii) We construct a time dependent VAE and show the effects of the auto-pruning in this model. As a result, we are able to model financial data sequences and estimate the value-at-risk (VaR) of associated portfolios. Our results show that we surpass the standard benchmarks for VaR estimation.

Risk management is an indispensable component of the financial system. In this context, capital requirements are built by financial institutions to avoid future bankruptcy. Their calculation is based on a specific kind of maps, so-called risk measures. There exist several forms and definitions of them. Multi-asset risk measures are the starting point of this dissertation. They determine the capital requirements as the minimal amount of money invested into multiple eligible assets to secure future payoffs. The dissertation consists of three main contributions: First, multi-asset risk measures are used to calculate pricing bounds for European type options. Second, multi-asset risk measures are combined with recently proposed intrinsic risk measures to obtain a new kind of a risk measure which we call a multi-asset intrinsic (MAI) risk measure. Third, the preferences of an agent are included in the calculation of the capital requirements. This leads to another new risk measure which we call a scalarized utility-based multi-asset (SUBMA) risk measure. In the introductory chapter, we recall the definition and properties of multi-asset risk measures. Then, each of the aforementioned contributions covers a separate chapter. In the following, the content of these three chapters is explained in more detail: Risk measures can be used to calculate pricing bounds for financial derivatives. In Chapter 2, we deal with the pricing of European options in an incomplete financial market model. We use the common risk measures Value-at-Risk and Expected Shortfall to define good deals on a financial market with log-normally distributed rates of return. We show that the pricing bounds obtained from Value-at-Risk may have a non-smooth behavior under parameter changes. Additionally, we find situations in which the seller's bound for a call option is smaller than the buyer's bound. We identify the missing convexity of the Value-at-Risk as main reason for this behavior. Due to the strong connection between the obtained pricing bounds and the theory of risk measures, we further obtain new insights in the finiteness and the continuity of multi-asset risk measures. In Chapter 3, we construct the MAI risk measure. Therefore, recall that a multi-asset risk measure describes the minimal external capital that has to be raised into multiple eligible assets to make a future financial position acceptable, i.e., that it passes a capital adequacy test. Recently, the alternative methodology of intrinsic risk measures was introduced in the literature. These ask for the minimal proportion of the financial position that has to be reallocated to pass the capital adequacy test, i.e., only internal capital is used. We combine these two concepts and call this new type of risk measure an MAI risk measure. It allows to secure the financial position by external capital as well as reallocating parts of the portfolio as an internal rebooking. We investigate several properties to demonstrate similarities and differences to the two aforementioned classical types of risk measures. We find out that diversification reduces the capital requirement only in special situations depending on the financial positions. With the help of Sion's minimax theorem we also prove a dual representation for MAI risk measures. Finally, we determine capital requirements in a model motivated by the Solvency II methodology. In the final Chapter 4, we construct the SUBMA risk measure. In doing so, we consider the situation in which a financial institution has to satisfy a capital adequacy test, e.g., by the Basel Accords for banks or by Solvency II for insurers. If the financial situation of this institution is tight, then it can happen that no reallocation of the initial endowment would pass the capital adequacy test. The classical portfolio optimization approach breaks down and a capital increase is needed. We introduce the SUBMA risk measure which optimizes the hedging costs and the expected utility of the institution simultaneously subject to the capital adequacy test. We find out that the SUBMA risk measure is coherent if the utility function has constant relative risk aversion and the capital adequacy test leads to a coherent acceptance set. In a one-period financial market model we present a sufficient condition for the SUBMA risk measure to be finite-valued and continuous. Finally, we calculate the SUBMA risk measure in a continuous-time financial market model for two benchmark capital adequacy tests.

Die Felix-Klein-Modellierungswoche ist eine Projektwoche für Schülerinnen und Schüler der gymnasialen Oberstufe. Wir beschreiben hier die Projekte, die während der Modellierungswoche im Februar 2022 durchgeführt wurden. Der Themenschwerpunkt der Veranstaltung lautete "Winter".

The great interest in robust covering problems is manifold, especially due to the plenitude of real world applications and the additional incorporation of uncertainties which are inherent in many practical issues. In this thesis, for a fixed positive integer \(q\), we introduce and elaborate on a new robust covering problem, called Robust Min-\(q\)-Multiset-Multicover, and related problems. The common idea of these problems is, given a collection of subsets of a ground set, to decide on the frequency of choosing each subset so as to satisfy the uncertain demand of each overall occurring element. Yet, in contrast to general covering problems, the subsets may only cover at most \(q\) of their elements. Varying the properties of the occurring elements leads to a selection of four interesting robust covering problems which are investigated. We extensively analyze the complexity of the arising problems, also for various restrictions to particular classes of uncertainty sets. For a given problem, we either provide a polynomial time algorithm or show that, unless \(\text{P}=\text{NP}\), such an algorithm cannot exist. Furthermore, in the majority of cases, we even give evidence that a polynomial time approximation scheme is most likely not possible for the hard problem variants. Moreover, we aim for approximations and approximation algorithms for these hard variants, where we focus on Robust Min-\(q\)-Multiset-Multicover. For a wide class of uncertainty sets, we present the first known polynomial time approximation algorithm for Robust Min-\(q\)-Multiset-Multicover having a provable worst-case performance guarantee.

Im Projekt MAFoaM - Modular Algorithms for closed Foam Mechanics - des Fraunhofer ITWM in Zusammenarbeit mit dem Fraunhofer IMWS wurde eine Methode zur Analyse und Simulation geschlossenzelliger PMI-Hartschäume entwickelt. Die Zellstruktur der Hartschäume wurde auf Basis von CT-Aufnahmen modelliert, um ihr Verformungs- und Versagensverhalten zu simulieren, d.h. wie sich die Schäume unter Belastungen bis hin zum totalen Defekt verhalten. In der Diplomarbeit wird die bildanalytische Zellrekonstruktion für PMI-Hartschäume automatisiert. Die Zellrekonstruktion dient der Bestimmung von Mikrostrukturgrößen, also geometrischer Eigenschaften der Schaumzellen, wie z.B. Mittelwert und Varianz des Zellvolumens oder der Zelloberfläche.