## Fachbereich Mathematik

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#### Keywords

- Transaction Costs (2)
- Cohen-Lenstra heuristic (1)
- Consistent Price Processes (1)
- Gyroscopic (1)
- Image restoration (1)
- No-Arbitrage (1)
- Poisson noise (1)
- Portfolio Optimization (1)
- QVIs (1)
- Stochastic Control (1)
- Utility (1)
- diffusion models (1)
- image denoising (1)
- multiplicative noise (1)
- nonlocal filtering (1)
- probability distribution (1)
- real quadratic number fields (1)
- second class group (1)
- similarity measures (1)
- variational methods (1)

The main topic of this thesis is to define and analyze a multilevel Monte Carlo algorithm for path-dependent functionals of the solution of a stochastic differential equation (SDE) which is driven by a square integrable, \(d_X\)-dimensional Lévy process \(X\). We work with standard Lipschitz assumptions and denote by \(Y=(Y_t)_{t\in[0,1]}\) the \(d_Y\)-dimensional strong solution of the SDE.
We investigate the computation of expectations \(S(f) = \mathrm{E}[f(Y)]\) using randomized algorithms \(\widehat S\). Thereby, we are interested in the relation of the error and the computational cost of \(\widehat S\), where \(f:D[0,1] \to \mathbb{R}\) ranges in the class \(F\) of measurable functionals on the space of càdlàg functions on \([0,1]\), that are Lipschitz continuous with respect to the supremum norm.
We consider as error \(e(\widehat S)\) the worst case of the root mean square error over the class of functionals \(F\). The computational cost of an algorithm \(\widehat S\), denoted \(\mathrm{cost}(\widehat S)\), should represent the runtime of the algorithm on a computer. We work in the real number model of computation and further suppose that evaluations of \(f\) are possible for piecewise constant functions in time units according to its number of breakpoints.
We state strong error estimates for an approximate Euler scheme on a random time discretization. With this strong error estimates, the multilevel algorithm leads to upper bounds for the convergence order of the error with respect to the computational cost. The main results can be summarized in terms of the Blumenthal-Getoor index of the driving Lévy process, denoted by \(\beta\in[0,2]\). For \(\beta <1\) and no Brownian component present, we almost reach convergence order \(1/2\), which means, that there exists a sequence of multilevel algorithms \((\widehat S_n)_{n\in \mathbb{N}}\) with \(\mathrm{cost}(\widehat S_n) \leq n\) such that \( e(\widehat S_n) \precsim n^{-1/2}\). Here, by \( \precsim\), we denote a weak asymptotic upper bound, i.e. the inequality holds up to an unspecified positive constant. If \(X\) has a Brownian component, the order has an additional logarithmic term, in which case, we reach \( e(\widehat S_n) \precsim n^{-1/2} \, (\log(n))^{3/2}\).
For the special subclass of $Y$ being the Lévy process itself, we also provide a lower bound, which, up to a logarithmic term, recovers the order \(1/2\), i.e., neglecting logarithmic terms, the multilevel algorithm is order optimal for \( \beta <1\).
An empirical error analysis via numerical experiments matches the theoretical results and completes the analysis.

This thesis is devoted to furthering the tropical intersection theory as well as to applying the
developed theory to gain new insights about tropical moduli spaces.
We use piecewise polynomials to define tropical cocycles that generalise the notion of tropical Cartier divisors to higher codimensions, introduce an intersection product of cocycles with tropical cycles and use the connection to toric geometry to prove a Poincaré duality for certain cases. Our
main application of this Poincaré duality is the construction of intersection-theoretic fibres under a
large class of tropical morphisms.
We construct an intersection product of cycles on matroid varieties which are a natural
generalisation of tropicalisations of classical linear spaces and the local blocks of smooth tropical
varieties. The key ingredient is the ability to express a matroid variety contained in another matroid variety by a piecewise polynomial that is given in terms of the rank functions of the corresponding
matroids. In particular, this enables us to intersect cycles on the moduli spaces of n-marked abstract
rational curves. We also construct a pull-back of cycles along morphisms of smooth varieties, relate
pull-backs to tropical modifications and show that every cycle on a matroid variety is rationally
equivalent to its recession cycle and can be cut out by a cocycle.
Finally, we define families of smooth rational tropical curves over smooth varieties and construct a tropical fibre product in order to show that every morphism of a smooth variety to the moduli space of abstract rational tropical curves induces a family of curves over the domain of the morphism.
This leads to an alternative, inductive way of constructing moduli spaces of rational curves.

Filtering, Approximation and Portfolio Optimization for Shot-Noise Models and the Heston Model
(2012)

We consider a continuous time market model in which stock returns satisfy a stochastic differential equation with stochastic drift, e.g. following an Ornstein-Uhlenbeck process. The driving noise of the stock returns consists not only of Brownian motion but also of a jump part (shot noise or compound Poisson process). The investor's objective is to maximize expected utility of terminal wealth under partial information which means that the investor only observes stock prices but does not observe the drift process. Since the drift of the stock prices is unobservable, it has to be estimated using filtering techniques. E.g., if the drift follows an Ornstein-Uhlenbeck process and without
jump part, Kalman filtering can be applied and optimal strategies can be computed explicitly. Also in other cases, like for an underlying
Markov chain, finite-dimensional filters exist. But for certain jump processes (e.g. shot noise) or certain nonlinear drift dynamics explicit computations, based on discrete observations, are no longer possible or existence of finite dimensional filters is no longer valid. The same
computational difficulties apply to the optimal strategy since it depends on the filter. In this case the model may be approximated by
a model where the filter is known and can be computed. E.g., we use statistical linearization for non-linear drift processes, finite-state-Markov chain approximations for the drift process and/or diffusion approximations for small jumps in the noise term.
In the approximating models, filters and optimal strategies can often be computed explicitly. We analyze and compare different approximation methods, in particular in view of performance of the corresponding utility maximizing strategies.

In this thesis we consider the problem of maximizing the growth rate with proportional and fixed costs in a framework with one bond and one stock, which is modeled as a jump diffusion with compound Poisson jumps. Following the approach from [1], we prove that in this framework it is optimal for an investor to follow a CB-strategy. The boundaries depend only on the parameters of the underlying stock and bond. Now it is natural to ask for the investor who follows a CB-strategy which is given by the stopping times \((\tau_i)_{i\in\mathbb N}\) and impulses \((\eta_i)_{i\in\mathbb N}\) how often he has to rebalance. In other words we want to obtain the limit of the inter trading times
\[
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n(\tau_{i+1}-\tau_{i}).
\]
We are able to obtain this limit which is given by the expected first exit time of the risky fraction process from some interval under the invariant measure of the Markov chain \((\eta_i)_{i\in\mathbb N}\) using the Ergodic Theorem from von Neumann and Birkhoff. In general, it is difficult to obtain the expectation of the first exit time for the process with jumps. Because of the jump part, when the process crosses the boundaries of the interval an overshoot may occur which makes it difficult to obtain the distribution. Nevertheless we can obtain the first exit time if the process has only negative jumps using scale functions. The main difficulty of this approach is that the scale functions are known only up to their Laplace transforms. In [2] and [3] the closed-form expression for the scale function of the Levy process with phase-type distributed jumps is obtained. Phase-type distributions build a rich class of positive-valued distributions: the exponential, hyperexponential, Erlang, hyper-Erlang and Coxian distributions. Since the scale function is given as a function in a closed form we can differentiate to obtain the expected first exit time using the fluctuation identities explicitly.
[1] Irle, A. and Sass,J.: Optimal portfolio policies under fixed and proportional transaction costs, Advances in Applied Probability 38, 916-942.
[2] Egami, M., Yamazaki, K.: On scale functions of spectrally negative Levy processes with phase-type jumps, working paper, July 3.
[3]Egami, M., Yamazaki, K.: Precautionary measures for credit risk management in jump models, working paper, June 17.

The goal of this work is to develop a simulation-based algorithm, allowing the prediction
of the effective mechanical properties of textiles on the basis of their microstructure
and corresponding properties of fibers. This method can be used for optimization of the
microstructure, in order to obtain a better stiffness or strength of the corresponding fiber
material later on. An additional aspect of the thesis is that we want to take into account the microcontacts
between fibers of the textile. One more aspect of the thesis is the accounting for the thickness of thin fibers in the
textile. An introduction of an additional asymptotics with respect to a small parameter,
the relation between the thickness and the representative length of the fibers, allows a
reduction of local contact problems between fibers to 1-dimensional problems, which
reduces numerical computations significantly.
A fiber composite material with periodic microstructure and multiple frictional microcontacts
between fibers is studied. The textile is modeled by introducing small geometrical
parameters: the periodicity of the microstructure and the characteristic
diameter of fibers. The contact linear elasticity problem is considered. A two-scale
approach is used for obtaining the effective mechanical properties.
The algorithm using asymptotic two-scale homogenization for computation of the
effective mechanical properties of textiles with periodic rod or fiber microstructure
is proposed. The algorithm is based on the consequent passing to the asymptotics
with respect to the in-plane period and the characteristic diameter of fibers. This
allows to come to the equivalent homogenized problem and to reduce the dimension
of the auxiliary problems. Further numerical simulations of the cell problems give
the effective material properties of the textile.
The homogenization of the boundary conditions on the vanishing out-of-plane interface
of a textile or fiber structured layer has been studied. Introducing additional
auxiliary functions into the formal asymptotic expansion for a heterogeneous
plate, the corresponding auxiliary and homogenized problems for a nonhomogeneous
Neumann boundary condition were deduced. It is incorporated into the right hand
side of the homogenized problem via effective out-of-plane moduli.
FiberFEM, a C++ finite element code for solving contact elasticity problems, is
developed. The code is based on the implementation of the algorithm for the contact
between fibers, proposed in the thesis.
Numerical examples of homogenization of geotexiles and wovens are obtained in the
work by implementation of the developed algorithm. The effective material moduli
are computed numerically using the finite element solutions of the auxiliary contact
problems obtained by FiberFEM.

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.

Image restoration and enhancement methods that respect important features such as edges play a fundamental role in digital image processing. In the last decades a large
variety of methods have been proposed. Nevertheless, the correct restoration and
preservation of, e.g., sharp corners, crossings or texture in images is still a challenge, in particular in the presence of severe distortions. Moreover, in the context of image denoising many methods are designed for the removal of additive Gaussian noise and their adaptation for other types of noise occurring in practice requires usually additional efforts.
The aim of this thesis is to contribute to these topics and to develop and analyze new
methods for restoring images corrupted by different types of noise:
First, we present variational models and diffusion methods which are particularly well
suited for the restoration of sharp corners and X junctions in images corrupted by
strong additive Gaussian noise. For their deduction we present and analyze different
tensor based methods for locally estimating orientations in images and show how to
successfully incorporate the obtained information in the denoising process. The advantageous
properties of the obtained methods are shown theoretically as well as by
numerical experiments. Moreover, the potential of the proposed methods is demonstrated
for applications beyond image denoising.
Afterwards, we focus on variational methods for the restoration of images corrupted
by Poisson and multiplicative Gamma noise. Here, different methods from the literature
are compared and the surprising equivalence between a standard model for
the removal of Poisson noise and a recently introduced approach for multiplicative
Gamma noise is proven. Since this Poisson model has not been considered for multiplicative
Gamma noise before, we investigate its properties further for more general
regularizers including also nonlocal ones. Moreover, an efficient algorithm for solving
the involved minimization problems is proposed, which can also handle an additional
linear transformation of the data. The good performance of this algorithm is demonstrated
experimentally and different examples with images corrupted by Poisson and
multiplicative Gamma noise are presented.
In the final part of this thesis new nonlocal filters for images corrupted by multiplicative
noise are presented. These filters are deduced in a weighted maximum likelihood
estimation framework and for the definition of the involved weights a new similarity measure for the comparison of data corrupted by multiplicative noise is applied. The
advantageous properties of the new measure are demonstrated theoretically and by
numerical examples. Besides, denoising results for images corrupted by multiplicative
Gamma and Rayleigh noise show the very good performance of the new filters.

Diese Dissertation besteht aus zwei aktuellen Themen im Bereich Finanzmathematik, die voneinander unabhängig sind.
Beim ersten Thema, "Flexible Algorithmen zur Bewertung komplexer Optionen mit mehreren Eigenschaften mittels der funktionalen Programmiersprache Haskell", handelt es sich um ein interdisziplinäres Projekt, in dem eine wissenschaftliche Brücke zwischen der Optionsbewertung und der funktionalen Programmierung geschlagen wurde.
Im diesem Projekt wurde eine funktionale Bibliothek zur Konstruktion von Optionen
entworfen, in dem es eine Reihe von grundlegenden Konstruktoren gibt, mit denen
man verschiedene Optionen kombinieren kann. Im Rahmen der funktionalen Bibliothek
wurde ein allgemeiner Algorithmus entwickelt, durch den die aus den Konstruktoren
kombinierten Optionen bewertet werden können.
Der mathematische Aspekt des Projekts besteht in der Entwicklung eines neuen Konzeptes zur Bewertung der Optionen. Dieses Konzept basiert auf dem Binomialmodell, welches in den letzten Jahren eine weite Verbreitung im Forschungsgebiet der Optionsbewertung fand. Der kerne Algorithmus des Konzeptes ist eine Kombination von mehreren
sorgfältig ausgewählten numerischen Methoden in Bezug auf den Binomialbaum. Diese
Kombination ist nicht trivial, sondern entwikelt sich nach bestimmten Regeln und ist eng mit den grundlegenden Konstruktoren verknüpft.
Ein wichtiger Charakterzug des Projekts ist die funktionale Denkweise. D. h. der Algorithmus ließ sich mithilfe einer funktionalen Programmiersprache formulieren. In unserem Projekt wurde Haskell verwendet.
Das zweite Thema, Monte-Carlo-Simulation des Deltas und (Cross-)Gammas von
Bermuda-Swaptions im LIBOR-Marktmodell, bezieht sich auf ein zentrales Problem der
Finanzmathematik, nämlich die Bestimmung der Risikoparameter komplexer Zinsderivate.
In dieser Arbeit wurde die numerische Berechnung des Delta-Vektors einer Bermuda-
Swaption ausführlich untersucht und die neue Herausforderung, die Gamma-Matrix einer Bermuda-Swaption exakt simulieren, erfolgreich gemeistert. Die beiden Risikoparameter spielen bei Handelsstrategien in Form des Delta-Hedgings und Gamma-Hedgings eine entscheidende Rolle. Das zugrunde liegende Zinsstrukturmodell ist das LIBORMarktmodell, welches in den letzten Jahren eine auffällige Entwicklung in der Finanzmathematik gemacht hat. Bei der Simulation und Anwendung des LIBOR-Marktmodells fällt die Monte-Carlo-Simulation ins Gewicht.
Für die Berechung des Delta-Vektors einer Bermuda-Swaption wurden drei klassische und drei von uns entwickelte numerische Methoden vorgestellt und gegenübergestellt, welche fast alle vorhandenen Arten der Monte-Carlo-Simulation zur Berechnung des Delta-Vektors einer Bermuda-Swaption enthalten.
Darüber hinaus gibt es in der Arbeit noch zwei neu entwickelte Methoden, um die Gamma-Matrix einer Bermuda-Swaption exakt zu berechnen, was völlig neu im Forschungsgebiet der Computational-Finance ist. Eine ist die modifizierte Finite-Differenzen-Methode. Die andere ist die reine Pathwise-Methode, die auf pfadweiser Differentialrechnung basiert und einem robusten und erwartungstreuen Simulationsverfahren entspricht.

Paper production is a problem with significant importance for the society and it is a challenging topic for scientific investigations. This study is concerned with the simulations of the pressing section of a paper machine. We aim at the development of an advanced mathematical model of the pressing section, which is able to recover the behavior of the fluid flow within the paper felt sandwich obtained in laboratory experiments.
From the modeling point of view the pressing of the paper-felt sandwich is a complex process since one has to deal with the two-phase flow in moving and deformable porous media. To account for the solid deformations, we use developments from the PhD thesis by S. Rief where the elasticity model is stated and discussed in detail. The flow model which accounts for the movement of water within the paper-felt sandwich is described with the help of two flow regimes: single-phase water flow and two-phase air-water flow. The model for the saturated flow is presented by the Darcy's law and the mass conservation. The second regime is described by the Richards' approach together with dynamic capillary effects. The model for the dynamic capillary pressure - saturation relation proposed by Hassanizadeh and Gray is adapted for the needs of the paper manufacturing process.
We have started the development of the flow model with the mathematical modeling in one-dimensional case. The one-dimensional flow model is derived from a two-dimensional one by an averaging procedure in vertical direction. The model is numerically studied and verified in comparison with measurements. Some theoretical investigations are performed to prove the convergence of the discrete solution to the continuous one. For completeness of the studies, the models with the static and dynamic capillary pressure–saturation relations are considered. Existence, compactness and convergence results are obtained for both models.
Then, a two-dimensional model is developed, which accounts for a multilayer computational domain and formation of the fully saturated zones. For discretization we use a non-orthogonal grid resolving the layer interfaces and the multipoint flux approximation O-method. The numerical experiments are carried out for parameters which are typical for the production process. The static and dynamic capillary pressure-saturation relations are tested to evaluate the influence of the dynamic capillary effect.
The last part of the thesis is an investigation of the validity range of the Richards’ assumption for the two-dimensional flow model with the static capillary pressure-saturation relation. Numerical experiments show that the Richards’ assumption is not the best choice in simulating processes in the pressing section.

Standard bases are one of the main tools in computational commutative algebra. In 1965
Buchberger presented a criterion for such bases and thus was able to introduce a first approach for their computation. Since the basic version of this algorithm is rather inefficient
due to the fact that it processes lots of useless data during its execution, active research for
improvements of those kind of algorithms is quite important.
In this thesis we introduce the reader to the area of computational commutative algebra with a focus on so-called signature-based standard basis algorithms. We do not only
present the basic version of Buchberger’s algorithm, but give an extensive discussion of different attempts optimizing standard basis computations, from several sorting algorithms
for internal data up to different reduction processes. Afterwards the reader gets a complete
introduction to the origin of signature-based algorithms in general, explaining the under-
lying ideas in detail. Furthermore, we give an extensive discussion in terms of correctness,
termination, and efficiency, presenting various different variants of signature-based standard basis algorithms.
Whereas Buchberger and others found criteria to discard useless computations which
are completely based on the polynomial structure of the elements considered, Faugère presented a first signature-based algorithm in 2002, the F5 Algorithm. This algorithm is famous for generating much less computational overhead during its execution. Within this
thesis we not only present Faugère’s ideas, we also generalize them and end up with several
different, optimized variants of his criteria for detecting redundant data.
Being not completely focussed on theory, we also present information about practical
aspects, comparing the performance of various implementations of those algorithms in the
computer algebra system Singular over a wide range of example sets.
In the end we give a rather extensive overview of recent research in this area of computational commutative algebra.