## Fachbereich Mathematik

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- Manifolds (2017)
- Lecture notes written to accompany a one semester course introducing to differential manifolds. Beyond the basic notions differential forms including Stokes' theorem are treated, as well as vector fields and flows on a differential manifold.

- Having a Plan B for Robust Optimization (2017)
- We extend the standard concept of robust optimization by the introduction of an alternative solution. In contrast to the classic concept, one is allowed to chose two solutions from which the best can be picked after the uncertain scenario has been revealed. We focus in this paper on the resulting robust problem for combinatorial problems with bounded uncertainty sets. We present a reformulation of the robust problem which decomposes it into polynomially many subproblems. In each subproblem one needs to find two solutions which are connected by a cost function which penalizes if the same element is part of both solutions. Using this reformulation, we show how the robust problem can be solved efficiently for the unconstrained combinatorial problem, the selection problem, and the minimum spanning tree problem. The robust problem corresponding to the shortest path problem turns out to be NP-complete on general graphs. However, for series-parallel graphs, the robust shortest path problem can be solved efficiently. Further, we show how approximation algorithms for the subproblem can be used to compute approximate solutions for the original problem.

- Portfolio Optimization with Risk Constraints in the View of Stochastic Interest Rates (2017)
- We discuss the portfolio selection problem of an investor/portfolio manager in an arbitrage-free financial market where a money market account, coupon bonds and a stock are traded continuously. We allow for stochastic interest rates and in particular consider one and two-factor Vasicek models for the instantaneous short rates. In both cases we consider a complete and an incomplete market setting by adding a suitable number of bonds. The goal of an investor is to find a portfolio which maximizes expected utility from terminal wealth under budget and present expected short-fall (PESF) risk constraints. We analyze this portfolio optimization problem in both complete and incomplete financial markets in three different cases: (a) when the PESF risk is minimum, (b) when the PESF risk is between minimum and maximum and (c) without risk constraints. (a) corresponds to the portfolio insurer problem, in (b) the risk constraint is binding, i.e., it is satisfied with equality, and (c) corresponds to the unconstrained Merton investment. In all cases we find the optimal terminal wealth and portfolio process using the martingale method and Malliavin calculus respectively. In particular we solve in the incomplete market settings the dual problem explicitly. We compare the optimal terminal wealth in the cases mentioned using numerical examples. Without risk constraints, we further compare the investment strategies for complete and incomplete market numerically.

- Asymptotics for change-point tests and change-point estimators (2017)
- In change-point analysis the point of interest is to decide if the observations follow one model or if there is at least one time-point, where the model has changed. This results in two sub- fields, the testing of a change and the estimation of the time of change. This thesis considers both parts but with the restriction of testing and estimating for at most one change-point. A well known example is based on independent observations having one change in the mean. Based on the likelihood ratio test a test statistic with an asymptotic Gumbel distribution was derived for this model. As it is a well-known fact that the corresponding convergence rate is very slow, modifications of the test using a weight function were considered. Those tests have a better performance. We focus on this class of test statistics. The first part gives a detailed introduction to the techniques for analysing test statistics and estimators. Therefore we consider the multivariate mean change model and focus on the effects of the weight function. In the case of change-point estimators we can distinguish between the assumption of a fixed size of change (fixed alternative) and the assumption that the size of the change is converging to 0 (local alternative). Especially, the fixed case in rarely analysed in the literature. We show how to come from the proof for the fixed alternative to the proof of the local alternative. Finally, we give a simulation study for heavy tailed multivariate observations. The main part of this thesis focuses on two points. First, analysing test statistics and, secondly, analysing the corresponding change-point estimators. In both cases, we first consider a change in the mean for independent observations but relaxing the moment condition. Based on a robust estimator for the mean, we derive a new type of change-point test having a randomized weight function. Secondly, we analyse non-linear autoregressive models with unknown regression function. Based on neural networks, test statistics and estimators are derived for correctly specified as well as for misspecified situations. This part extends the literature as we analyse test statistics and estimators not only based on the sample residuals. In both sections, the section on tests and the one on the change-point estimator, we end with giving regularity conditions on the model as well as the parameter estimator. Finally, a simulation study for the case of the neural network based test and estimator is given. We discuss the behaviour under correct and mis-specification and apply the neural network based test and estimator on two data sets.

- Small self-centralizing subgroups in defect groups of finite classical groups (2017)
- In this thesis, we consider a problem from modular representation theory of finite groups. Lluís Puig asked the question whether the order of the defect groups of a block \( B \) of the group algebra of a given finite group \( G \) can always be bounded in terms of the order of the vertices of an arbitrary simple module lying in \( B \). In characteristic \( 2 \), there are examples showing that this is not possible in general, whereas in odd characteristic, no such examples are known. For instance, it is known that the answer to Puig's question is positive in case that \( G \) is a symmetric group, by work of Danz, Külshammer, and Puig. Motivated by this, we study the cases where \( G \) is a finite classical group in non-defining characteristic or one of the finite groups \( G_2(q) \) or \( ³D_4(q) \) of Lie type, again in non-defining characteristic. Here, we generalize Puig's original question by replacing the vertices occurring in his question by arbitrary self-centralizing subgroups of the defect groups. We derive positive and negative answers to this generalized question. \[\] In addition to that, we determine the vertices of the unipotent simple \( GL_2(q) \)-module labeled by the partition \( (1,1) \) in characteristic \( 2 \). This is done using a method known as Brauer construction.

- Convex Analysis for Processing Hyperspectral Images and Data from Hadamard Spaces (2017)
- This thesis brings together convex analysis and hyperspectral image processing. Convex analysis is the study of convex functions and their properties. Convex functions are important because they admit minimization by efficient algorithms and the solution of many optimization problems can be formulated as minimization of a convex objective function, extending much beyond the classical image restoration problems of denoising, deblurring and inpainting. \(\hspace{1mm}\) At the heart of convex analysis is the duality mapping induced within the class of convex functions by the Fenchel transform. In the last decades efficient optimization algorithms have been developed based on the Fenchel transform and the concept of infimal convolution. \(\hspace{1mm}\) The infimal convolution is of similar importance in convex analysis as the convolution in classical analysis. In particular, the infimal convolution with scaled parabolas gives rise to the one parameter family of Moreau-Yosida envelopes, which approximate a given function from below while preserving its minimum value and minimizers. The closely related proximal mapping replaces the gradient step in a recently developed class of efficient first-order iterative minimization algorithms for non-differentiable functions. For a finite convex function, the proximal mapping coincides with a gradient step of its Moreau-Yosida envelope. Efficient algorithms are needed in hyperspectral image processing, where several hundred intensity values measured in each spatial point give rise to large data volumes. \(\hspace{1mm}\) In the \(\textbf{first part}\) of this thesis, we are concerned with models and algorithms for hyperspectral unmixing. As part of this thesis a hyperspectral imaging system was taken into operation at the Fraunhofer ITWM Kaiserslautern to evaluate the developed algorithms on real data. Motivated by missing-pixel defects common in current hyperspectral imaging systems, we propose a total variation regularized unmixing model for incomplete and noisy data for the case when pure spectra are given. We minimize the proposed model by a primal-dual algorithm based on the proximum mapping and the Fenchel transform. To solve the unmixing problem when only a library of pure spectra is provided, we study a modification which includes a sparsity regularizer into model. \(\hspace{1mm}\) We end the first part with the convergence analysis for a multiplicative algorithm derived by optimization transfer. The proposed algorithm extends well-known multiplicative update rules for minimizing the Kullback-Leibler divergence, to solve a hyperspectral unmixing model in the case when no prior knowledge of pure spectra is given. \(\hspace{1mm}\) In the \(\textbf{second part}\) of this thesis, we study the properties of Moreau-Yosida envelopes, first for functions defined on Hadamard manifolds, which are (possibly) infinite-dimensional Riemannian manifolds with negative curvature, and then for functions defined on Hadamard spaces. \(\hspace{1mm}\) In particular we extend to infinite-dimensional Riemannian manifolds an expression for the gradient of the Moreau-Yosida envelope in terms of the proximal mapping. With the help of this expression we show that a sequence of functions converges to a given limit function in the sense of Mosco if the corresponding Moreau-Yosida envelopes converge pointwise at all scales. \(\hspace{1mm}\) Finally we extend this result to the more general setting of Hadamard spaces. As the reverse implication is already known, this unites two definitions of Mosco convergence on Hadamard spaces, which have both been used in the literature, and whose equivalence has not yet been known.

- Product Pricing with Additive Influences - Algorithms and Complexity Results for Pricing in Social Networks (2017)
- We introduce and investigate a product pricing model in social networks where the value a possible buyer assigns to a product is influenced by the previous buyers. The selling proceeds in discrete, synchronous rounds for some set price and the individual values are additively altered. Whereas computing the revenue for a given price can be done in polynomial time, we show that the basic problem PPAI, i.e., is there a price generating a requested revenue, is weakly NP-complete. With algorithm Frag we provide a pseudo-polynomial time algorithm checking the range of prices in intervals of common buying behavior we call fragments. In some special cases, e.g., solely positive influences, graphs with bounded in-degree, or graphs with bounded path length, the amount of fragments is polynomial. Since the run-time of Frag is polynomial in the amount of fragments, the algorithm itself is polynomial for these special cases. For graphs with positive influence we show that every buyer does also buy for lower prices, a property that is not inherent for arbitrary graphs. Algorithm FixHighest improves the run-time on these graphs by using the above property. Furthermore, we introduce variations on this basic model. The version of delaying the propagation of influences and the awareness of the product can be implemented in our basic model by substituting nodes and arcs with simple gadgets. In the chapter on Dynamic Product Pricing we allow price changes, thereby raising the complexity even for graphs with solely positive or negative influences. Concerning Perishable Product Pricing, i.e., the selling of products that are usable for some time and can be rebought afterward, the principal problem is computing the revenue that a given price can generate in some time horizon. In general, the problem is #P-hard and algorithm Break runs in pseudo-polynomial time. For polynomially computable revenue, we investigate once more the complexity to find the best price. We conclude the thesis with short results in topics of Cooperative Pricing, Initial Value as Parameter, Two Product Pricing, and Bounded Additive Influence.

- Mathematical modelling of interacting fibre structures and non-woven materials (2017)
- Non–woven materials consist of many thousands of fibres laid down on a conveyor belt under the influence of a turbulent air stream. To improve industrial processes for the production of non–woven materials, we develop and explore novel mathematical fibre and material models. In Part I of this thesis we improve existing mathematical models describing the fibres on the belt in the meltspinning process. In contrast to existing models, we include the fibre–fibre interaction caused by the fibres’ thickness which prevents the intersection of the fibres and, hence, results in a more accurate mathematical description. We start from a microscopic characterisation, where each fibre is described by a stochastic functional differential equation and include the interaction along the whole fibre path, which is described by a delay term. As many fibres are required for the production of a non–woven material, we consider the corresponding mean–field equation, which describes the evolution of the fibre distribution with respect to fibre position and orientation. To analyse the particular case of large turbulences in the air stream, we develop the diffusion approximation which yields a distribution describing the fibre position. Considering the convergence to equilibrium on an analytical level, as well as performing numerical experiments, gives an insight into the influence of the novel interaction term in the equations. In Part II of this thesis we model the industrial airlay process, which is a production method whereby many short fibres build a three–dimensional non–woven material. We focus on the development of a material model based on original fibre properties, machine data and micro computer tomography. A possible linking of these models to other simulation tools, for example virtual tensile tests, is discussed. The models and methods presented in this thesis promise to further the field in mathematical modelling and computational simulation of non–woven materials.

- The Split tree for option pricing (2017)
- In this dissertation convergence of binomial trees for option pricing is investigated. The focus is on American and European put and call options. For that purpose variations of the binomial tree model are reviewed. In the first part of the thesis we investigated the convergence behavior of the already known trees from the literature (CRR, RB, Tian and CP) for the European options. The CRR and the RB tree suffer from irregular convergence, so our first aim is to find a way to get the smooth convergence. We first show what causes these oscillations. That will also help us to improve the rate of convergence. As a result we introduce the Tian and the CP tree and we proved that the order of convergence for these trees is \(O \left(\frac{1}{n} \right)\). Afterwards we introduce the Split tree and explain its properties. We prove the convergence of it and we found an explicit first order error formula. In our setting, the splitting time \(t_{k} = k\Delta t\) is not fixed, i.e. it can be any time between 0 and the maturity time \(T\). This is the main difference compared to the model from the literature. Namely, we show that the good properties of the CRR tree when \(S_{0} = K\) can be preserved even without this condition (which is mainly the case). We achieved the convergence of \(O \left(n^{-\frac{3}{2}} \right)\) and we typically get better results if we split our tree later.

- Continuous-Time Portfolio Optimization under Partial Information and Convex Constraints: Deriving Explicit Results (2017)
- In this thesis we explicitly solve several portfolio optimization problems in a very realistic setting. The fundamental assumptions on the market setting are motivated by practical experience and the resulting optimal strategies are challenged in numerical simulations. We consider an investor who wants to maximize expected utility of terminal wealth by trading in a high-dimensional financial market with one riskless asset and several stocks. The stock returns are driven by a Brownian motion and their drift is modelled by a Gaussian random variable. We consider a partial information setting, where the drift is unknown to the investor and has to be estimated from the observable stock prices in addition to some analyst’s opinion as proposed in [CLMZ06]. The best estimate given these observations is the well known Kalman-Bucy-Filter. We then consider an innovations process to transform the partial information setting into a market with complete information and an observable Gaussian drift process. The investor is restricted to portfolio strategies satisfying several convex constraints. These constraints can be due to legal restrictions, due to fund design or due to client's specifications. We cover in particular no-short-selling and no-borrowing constraints. One popular approach to constrained portfolio optimization is the convex duality approach of Cvitanic and Karatzas. In [CK92] they introduce auxiliary stock markets with shifted market parameters and obtain a dual problem to the original portfolio optimization problem that can be better solvable than the primal problem. Hence we consider this duality approach and using stochastic control methods we first solve the dual problems in the cases of logarithmic and power utility. Here we apply a reverse separation approach in order to obtain areas where the corresponding Hamilton-Jacobi-Bellman differential equation can be solved. It turns out that these areas have a straightforward interpretation in terms of the resulting portfolio strategy. The areas differ between active and passive stocks, where active stocks are invested in, while passive stocks are not. Afterwards we solve the auxiliary market given the optimal dual processes in a more general setting, allowing for various market settings and various dual processes. We obtain explicit analytical formulas for the optimal portfolio policies and provide an algorithm that determines the correct formula for the optimal strategy in any case. We also show optimality of our resulting portfolio strategies in different verification theorems. Subsequently we challenge our theoretical results in a historical and an artificial simulation that are even closer to the real world market than the setting we used to derive our theoretical results. However, we still obtain compelling results indicating that our optimal strategies can outperform any benchmark in a real market in general.