### Refine

#### Year of publication

#### Document Type

- Preprint (605)
- Doctoral Thesis (237)
- Report (121)
- Article (34)
- Diploma Thesis (25)
- Lecture (22)
- Master's Thesis (6)
- Part of a Book (4)
- Study Thesis (4)
- Working Paper (4)

#### Keywords

- Wavelet (14)
- Inverses Problem (12)
- Mehrskalenanalyse (12)
- Modellierung (12)
- Mathematikunterricht (9)
- Mathematische Modellierung (9)
- praxisorientiert (9)
- Approximation (8)
- Boltzmann Equation (8)
- MINT (8)

#### Faculty / Organisational entity

- Fachbereich Mathematik (1067) (remove)

Einführung in die Algebra
(2020)

Synapses are connections between different nerve cells that form an essential link in neural signal transmission. It is generally distinguished between electrical and chemical synapses, where chemical synapses are more common in the human brain and are also the type we deal with in this work.
In chemical synapses, small container-like objects called vesicles fill with neurotransmitter and expel them from the cell during synaptic transmission. This process is vital for communication between neurons. However, to the best of our knowledge no mathematical models that take different filling states of the vesicles into account have been developed before this thesis was written.
In this thesis we propose a novel mathematical model for modeling synaptic transmission at chemical synapses which includes the description of vesicles of different filling states. The model consists of a transport equation (for the vesicle growth process) plus three ordinary differential equations (ODEs) and focuses on the presynapse and synaptic cleft.
The well-posedness is proved in detail for this partial differential equation (PDE) system. We also propose a few different variations and related models. In particular, an ODE system is derived and a delay differential equation (DDE) system is formulated. We then use nonlinear optimization methods for data fitting to test some of the models on data made available to us by the Animal Physiology group at TU Kaiserslautern.

Elementare Zahlentheorie
(2020)

In a recent paper, G. Malle and G. Robinson proposed a modular anologue to Brauer's famous \( k(B) \)-conjecture. If \( B \) is a \( p \)-block of a finite group with defect group \( D \), then they conjecture that \( l(B) \leq p^r \), where \( r \) is the sectional \( p \)-rank of \( D \). Since this conjecture is relatively new, there is obviously still a lot of work to do. This thesis is concerned with proving their conjecture for the finite groups of exceptional Lie type.

Diversification is one of the main pillars of investment strategies. The prominent 1/N portfolio, which puts equal weight on each asset is, apart from its simplicity, a method which is hard to outperform in realistic settings, as many studies have shown. However, depending on the number of considered assets, this method can lead to very large portfolios. On the other hand, optimization methods like the mean-variance portfolio suffer from estimation errors, which often destroy the theoretical benefits. We investigate the performance of the equal weight portfolio when using fewer assets. For this we explore different naive portfolios, from selecting the best Sharpe ratio assets to exploiting knowledge about correlation structures using clustering methods. The clustering techniques separate the possible assets into non-overlapping clusters and the assets within a cluster are ordered by their Sharpe ratio. Then the best asset of each portfolio is chosen to be a member of the new portfolio with equal weights, the cluster portfolio. We show that this portfolio inherits the advantages of the 1/N portfolio and can even outperform it empirically. For this we use real data and several simulation models. We prove these findings from a statistical point of view using the framework by DeMiguel, Garlappi and Uppal (2009). Moreover, we show the superiority regarding the Sharpe ratio in a setting, where in each cluster the assets are comonotonic. In addition, we recommend the consideration of a diversification-risk ratio to evaluate the performance of different portfolios.

LinTim is a scientific software toolbox that has been under development since 2007, giving the possibility to solve the various planning steps in public transportation. Although the name originally derives from "Lineplanning and Timetabling", the available functions have grown far beyond this scope.
This document is the documentation for version 2020.02.
For more information, see https://www.lintim.net

A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations (PDEs) and switched differential algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modeled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.

In this thesis we study a variant of the quadrature problem for stochastic differential equations (SDEs), namely the approximation of expectations \(\mathrm{E}(f(X))\), where \(X = (X(t))_{t \in [0,1]}\) is the solution of an SDE and \(f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R}\) is a functional, mapping each realization of \(X\) into the real numbers. The distinctive feature in this work is that we consider randomized (Monte Carlo) algorithms with random bits as their only source of randomness, whereas the algorithms commonly studied in the literature are allowed to sample from the uniform distribution on the unit interval, i.e., they do have access to random numbers from \([0,1]\).
By assumption, all further operations like, e.g., arithmetic operations, evaluations of elementary functions, and oracle calls to evaluate \(f\) are considered within the real number model of computation, i.e., they are carried out exactly.
In the following, we provide a detailed description of the quadrature problem, namely we are interested in the approximation of
\begin{align*}
S(f) = \mathrm{E}(f(X))
\end{align*}
for \(X\) being the \(r\)-dimensional solution of an autonomous SDE of the form
\begin{align*}
\mathrm{d}X(t) = a(X(t)) \, \mathrm{d}t + b(X(t)) \, \mathrm{d}W(t), \quad t \in [0,1],
\end{align*}
with deterministic initial value
\begin{align*}
X(0) = x_0 \in \mathbb{R}^r,
\end{align*}
and driven by a \(d\)-dimensional standard Brownian motion \(W\). Furthermore, the drift coefficient \(a \colon \mathbb{R}^r \to \mathbb{R}^r\) and the diffusion coefficient \(b \colon \mathbb{R}^r \to \mathbb{R}^{r \times d}\) are assumed to be globally Lipschitz continuous.
For the function classes
\begin{align*}
F_{\infty} = \bigl\{f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R} \colon |f(x) - f(y)| \leq \|x-y\|_{\sup}\bigr\}
\end{align*}
and
\begin{align*}
F_p = \bigl\{f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R} \colon |f(x) - f(y)| \leq \|x-y\|_{L_p}\bigr\}, \quad 1 \leq p < \infty.
\end{align*}
we have established the following.
\[\]
\(\textit{Theorem 1.}\)
There exists a random bit multilevel Monte Carlo (MLMC) algorithm \(M\) using
\[
L = L(\varepsilon,F) = \begin{cases}\lceil{\log_2(\varepsilon^{-2}}\rceil, &\text{if} \ F = F_p,\\
\lceil{\log_2(\varepsilon^{-2} + \log_2(\log_2(\varepsilon^{-1}))}\rceil, &\text{if} \ F = F_\infty
\end{cases}
\]
and replication numbers
\[
N_\ell = N_\ell(\varepsilon,F) = \begin{cases}
\lceil{(L+1) \cdot 2^{-\ell} \cdot \varepsilon^{-2}}\rceil, & \text{if} \ F = F_p,\\
\lceil{(L+1) \cdot 2^{-\ell} \cdot \max(\ell,1) \cdot \varepsilon^{-2}}\rceil, & \text{if} \ F=f_\infty
\end{cases}
\]
for \(\ell = 0,\ldots,L\), for which exists a positive constant \(c\) such that
\begin{align*}
\mathrm{error}(M,F) = \sup_{f \in F} \bigl(\mathrm{E}(S(f) - M(f))^2\bigr)^{1/2} \leq c \cdot \varepsilon
\end{align*}
and
\begin{align*}
\mathrm{cost}(M,F) = \sup_{f \in F} \mathrm{E}(\mathrm{cost}(M,f)) \leq c \cdot \varepsilon^{-2} \cdot \begin{cases}
(\ln(\varepsilon^{-1}))^2, &\text{if} \ F=F_p,\\
(\ln(\varepsilon^{-1}))^3, &\text{if} \ F=F_\infty
\end{cases}
\end{align*}
for every \(\varepsilon \in {]0,1/2[}\).
\[\]
Hence, in terms of the \(\varepsilon\)-complexity
\begin{align*}
\mathrm{comp}(\varepsilon,F) = \inf\bigl\{\mathrm{cost}(M,F) \colon M \ \text{is a random bit MC algorithm}, \mathrm{error}(M,F) \leq \varepsilon\bigr\}
\end{align*}
we have established the upper bound
\begin{align*}
\mathrm{comp}(\varepsilon,F) \leq c \cdot \varepsilon^{-2} \cdot \begin{cases}
(\ln(\varepsilon^{-1}))^2, &\text{if} \ F=F_p,\\
(\ln(\varepsilon^{-1}))^3, &\text{if} \ F=F_\infty
\end{cases}
\end{align*}
for some positive constant \(c\). That is, we have shown the same weak asymptotic upper bound as in the case of random numbers from \([0,1]\). Hence, in this sense, random bits are almost as powerful as random numbers for our computational problem.
Moreover, we present numerical results for a non-analyzed adaptive random bit MLMC Euler algorithm, in the particular cases of the Brownian motion, the geometric Brownian motion, the Ornstein-Uhlenbeck SDE and the Cox-Ingersoll-Ross SDE. We also provide a numerical comparison to the corresponding adaptive random number MLMC Euler method.
A key challenge in the analysis of the algorithm in Theorem 1 is the approximation of probability distributions by means of random bits. A problem very closely related to the quantization problem, i.e., the optimal approximation of a given probability measure (on a separable Hilbert space) by means of a probability measure with finite support size.
Though we have shown that the random bit approximation of the standard normal distribution is 'harder' than the corresponding quantization problem (lower weak rate of convergence), we have been able to establish the same weak rate of convergence as for the corresponding quantization problem in the case of the distribution of a Brownian bridge on \(L_2([0,1])\), the distribution of the solution of a scalar SDE on \(L_2([0,1])\), and the distribution of a centered Gaussian random element in a separable Hilbert space.

Model uncertainty is a challenge that is inherent in many applications of mathematical models in various areas, for instance in mathematical finance and stochastic control. Optimization procedures in general take place under a particular model. This model, however, might be misspecified due to statistical estimation errors and incomplete information. In that sense, any specified model must be understood as an approximation of the unknown "true" model. Difficulties arise since a strategy which is optimal under the approximating model might perform rather bad in the true model. A natural way to deal with model uncertainty is to consider worst-case optimization.
The optimization problems that we are interested in are utility maximization problems in continuous-time financial markets. It is well known that drift parameters in such markets are notoriously difficult to estimate. To obtain strategies that are robust with respect to a possible misspecification of the drift we consider a worst-case utility maximization problem with ellipsoidal uncertainty sets for the drift parameter and with a constraint on the strategies that prevents a pure bond investment.
By a dual approach we derive an explicit representation of the optimal strategy and prove a minimax theorem. This enables us to show that the optimal strategy converges to a generalized uniform diversification strategy as uncertainty increases.
To come up with a reasonable uncertainty set, investors can use filtering techniques to estimate the drift of asset returns based on return observations as well as external sources of information, so-called expert opinions. In a Black-Scholes type financial market with a Gaussian drift process we investigate the asymptotic behavior of the filter as the frequency of expert opinions tends to infinity. We derive limit theorems stating that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process which can be interpreted as a continuous-time expert. Our convergence results carry over to convergence of the value function in a portfolio optimization problem with logarithmic utility.
Lastly, we use our observations about how expert opinions improve drift estimates for our robust utility maximization problem. We show that our duality approach carries over to a financial market with non-constant drift and time-dependence in the uncertainty set. A time-dependent uncertainty set can then be defined based on a generic filter. We apply this to various investor filtrations and investigate which effect expert opinions have on the robust strategies.