Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics – serie

Jan Natolski behandelt die Problematik der Quantifizierung des Risikokapitals aus einer theoretischen Perspektive, die in wertvolle Impulse für die praktische Handhabung mündet. Dies ist ein wichtiger Schritt, da Versicherungsunternehmen durch die Richtlinie Solvency II verpflichtet sind, genügend Risikokapital zu hinterlegen, um die Gefahr der Insolvenz möglichst gering zu halten. Als zentrales Resultat zeigt der Autor, dass die in der Praxis verwendete Methode der Replikation mathematisch fundiert ist. Dabei setzt er Methoden aus verschiedenen mathematischen Gebieten, so z.B. der Optimierung und der Stochastik, ein.

Manuela Spangler deals with the default risk modelling of German covered bonds (Pfandbriefe). Existing credit risk models are not suitable for this purpose as they only consider the creditworthiness of the issuer while product-specific features are not taken into account. The author develops a multi-period simulation-based Pfandbrief model which adequately accounts for the product’s most important characteristics and risks. The model provides a flexible framework for structural analyses and can be easily extended for tailor-made investigations. While the focus of the work is on the specification of the model itself, simulation results from an exemplary model calibration are also discussed.About the Author Manuela Spangler works as a quantitative risk analyst for a large asset management company and holds a PhD in mathematics from the University of Augsburg. Prior to her current position, she worked as a risk manager and financial engineer in the banking and insurance sector for various years.

Emilia Graß develops a solution method which can provide fast and near-optimal solutions to realistic large-scale two-stage stochastic problems in disaster management. The author proposes a specialized interior-point method to accelerate the standard L-shaped algorithm. She shows that the newly developed solution method solves two realistic large-scale case studies for the hurricane prone Gulf and Atlantic coast faster than the standard L-shaped method and a commercial solver. The accelerated solution method enables relief organizations to employ appropriate preparation measures even in the case of short-term disaster warnings.About the AuthorEmilia Graß holds a PhD from the Hamburg University of Technology, Germany. She is currently working as guest researcher on the project cyber security in healthcare at the Centre for Health Policy, Imperial College London, UK. Her scientific focus is on stochastic programming, solution methods, disaster management and healthcare.

Ole Martin extends well-established techniques for the analysis of high-frequency data based on regular observations to the more general setting of asynchronous and irregular observations. Such methods are much needed in practice as real data usually comes in irregular form. In the theoretical part he develops laws of large numbers and central limit theorems as well as a new bootstrap procedure to assess asymptotic laws. The author then applies the theoretical results to estimate the quadratic covariation and to construct tests for the presence of common jumps. The simulation results show that in finite samples his methods despite the much more complex setting perform comparably well as methods based on regular data.​About the Author:Dr. Ole Martin completed his PhD at the Kiel University (CAU), Germany. His research focuses on high-frequency statistics for semimartingales with the aim to develop methods based on irregularlyobserved data.

Oleg Wilfer presents a new conjugate duality concept for geometric and cone constrained optimization problems whose objective functions are a composition of finitely many functions. As an application, the author derives results for single minmax location problems formulated by means of extended perturbed minimal time functions as well as for multi-facility minmax location problems defined by gauges. In addition, he provides formulae of projections onto the epigraphs of gauges to solve these kinds of location problems numerically by using parallel splitting algorithms. Numerical comparisons of recent methods show the excellent performance of the proposed solving technique.​About the Author:Dr. Oleg Wilfer received his PhD at the Faculty of Mathematics of Chemnitz University of Technology, Germany. He is currently working as a development engineer in the automotive industry.

Anja Schedel analyzes two models in the field of algorithmic game theory which both constitute bilevel problems in networks. The first model is a game-theoretic variant of the well-known Steiner forest problem, and one is interested in an optimal sharing of the cost of the Steiner forest. The author provides (and partially exactly characterizes) network structures which allow for cost-minimal pure Nash equilibria. The second model is motivated from privatized public roads, in which private, selfishly acting firms build roads, and as compensation for their investment, are allowed to set prices for using the roads. For a basic model of this situation, the author shows existence and uniqueness of pure Nash equilibria. The existence result requires a non-standard proof approach since techniques like Kakutani’s fixed point theorem cannot be applied directly.

Maximilian Klein analyses nested Monte Carlo simulations for the approximation of conditional expected values. Thereby, the book deals with two general risk functional classes for conditional expected values, on the one hand the class of moment-based estimators (notable examples are the probability of a large loss or the lower partial moments) and on the other hand the class of quantile-based estimators. For both functional classes, the almost sure convergence of the respective estimator is proven and the underlying convergence speed is quantified. In particular, the class of quantile-based estimators has important practical consequences especially for life insurance companies since the Value-at-Risk falls into this class and thus covers the solvency capital requirement problem. Furthermore, a novel non parametric confidence interval method for quantiles is presented which takes the additional noise of the inner simulation into account.

In this book Lukas Graf studies dynamic network flows which are a model for individual car traffic in road networks. It is assumed that drivers choose their routes based on information about the current state of the network in such a way as to selfishly minimize their own arrival time at their destination. Whilst on their journey the drivers adapt their current route choices based on the changing state of the network. A dynamic flow wherein every (infinitesimally small) flow particle behaves in this way is then called an instantaneous dynamic equilibrium. After giving a mathematically precise definition of this equilibrium concept the author shows existence of those equilibrium flows, studies their computational complexity and derives bounds on their quality.