## Fachbereich Mathematik

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- Fachbereich Mathematik (218)
- Fraunhofer (ITWM) (2)

In this thesis, we deal with the finite group of Lie type \(F_4(2^n)\). The aim is to find information on the \(l\)-decomposition numbers of \(F_4(2^n)\) on unipotent blocks for \(l\neq2\) and \(n\in \mathbb{N}\) arbitrary and on the irreducible characters of the Sylow \(2\)-subgroup of \(F_4(2^n)\).
S. M. Goodwin, T. Le, K. Magaard and A. Paolini have found a parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), a Sylow \(2\)-subgroup of \(F_4(q)\), of \(F_4(p^n)\), \(p\) a prime, for the case \(p\neq2\).
We managed to adapt their methods for the parametrization of the irreducible characters of the Sylow \(2\)-subgroup for the case \(p=2\) for the group \(F_4(q)\), \(q=p^n\). This gives a nearly complete parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), namely of all irreducible characters of \(U\) arising from so-called abelian cores.
The general strategy we have applied to obtain information about the \(l\)-decomposition numbers on unipotent blocks is to induce characters of the unipotent subgroup \(U\) of \(F_4(q)\) and Harish-Chandra induce projective characters of proper Levi subgroups of \(F_4(q)\) to obtain projective characters of \(F_4(q)\). Via Brauer reciprocity, the multiplicities of the ordinary irreducible unipotent characters in these projective characters give us information on the \(l\)-decomposition numbers of the unipotent characters of \(F_4(q)\).
Sadly, the projective characters of \(F_4(q)\) we obtained were not sufficient to give the shape of the entire decomposition matrix.

In this thesis we integrate discrete dividends into the stock model, estimate
future outstanding dividend payments and solve different portfolio optimization
problems. Therefore, we discuss three well-known stock models, including
discrete dividend payments and evolve a model, which also takes early
announcement into account.
In order to estimate the future outstanding dividend payments, we develop a
general estimation framework. First, we investigate a model-free, no-arbitrage
methodology, which is based on the put-call parity for European options. Our
approach integrates all available option market data and simultaneously calculates
the market-implied discount curve. We illustrate our method using stocks
of European blue-chip companies and show within a statistical assessment that
the estimate performs well in practice.
As American options are more common, we additionally develop a methodology,
which is based on market prices of American at-the-money options.
This method relies on a linear combination of no-arbitrage bounds of the dividends,
where the corresponding optimal weight is determined via a historical
least squares estimation using realized dividends. We demonstrate our method
using all Dow Jones Industrial Average constituents and provide a robustness
check with respect to the used discount factor. Furthermore, we backtest our
results against the method using European options and against a so called
simple estimate.
In the last part of the thesis we solve the terminal wealth portfolio optimization
problem for a dividend paying stock. In the case of the logarithmic utility
function, we show that the optimal strategy is not a constant anymore but
connected to the Merton strategy. Additionally, we solve a special optimal
consumption problem, where the investor is only allowed to consume dividends.
We show that this problem can be reduced to the before solved terminal wealth
problem.

In this thesis, we focus on the application of the Heath-Platen (HP) estimator in option
pricing. In particular, we extend the approach of the HP estimator for pricing path dependent
options under the Heston model. The theoretical background of the estimator
was first introduced by Heath and Platen [32]. The HP estimator was originally interpreted
as a control variate technique and an application for European vanilla options was
presented in [32]. For European vanilla options, the HP estimator provided a considerable
amount of variance reduction. Thus, applying the technique for path dependent options
under the Heston model is the main contribution of this thesis.
The first part of the thesis deals with the implementation of the HP estimator for pricing
one-sided knockout barrier options. The main difficulty for the implementation of the HP
estimator is located in the determination of the first hitting time of the barrier. To test the
efficiency of the HP estimator we conduct numerical tests with regard to various aspects.
We provide a comparison among the crude Monte Carlo estimation, the crude control
variate technique and the HP estimator for all types of barrier options. Furthermore, we
present the numerical results for at the money, in the money and out of the money barrier
options. As numerical results imply, the HP estimator performs superior among others
for pricing one-sided knockout barrier options under the Heston model.
Another contribution of this thesis is the application of the HP estimator in pricing bond
options under the Cox-Ingersoll-Ross (CIR) model and the Fong-Vasicek (FV) model. As
suggested in the original paper of Heath and Platen [32], the HP estimator has a wide
range of applicability for derivative pricing. Therefore, transferring the structure of the
HP estimator for pricing bond options is a promising contribution. As the approximating
Vasicek process does not seem to be as good as the deterministic volatility process in the
Heston setting, the performance of the HP estimator in the CIR model is only relatively
good. However, for the FV model the variance reduction provided by the HP estimator is
again considerable.
Finally, the numerical result concerning the weak convergence rate of the HP estimator
for pricing European vanilla options in the Heston model is presented. As supported by
numerical analysis, the HP estimator has weak convergence of order almost 1.

A popular model for the locations of fibres or grains in composite materials
is the inhomogeneous Poisson process in dimension 3. Its local intensity function
may be estimated non-parametrically by local smoothing, e.g. by kernel
estimates. They crucially depend on the choice of bandwidths as tuning parameters
controlling the smoothness of the resulting function estimate. In this
thesis, we propose a fast algorithm for learning suitable global and local bandwidths
from the data. It is well-known, that intensity estimation is closely
related to probability density estimation. As a by-product of our study, we
show that the difference is asymptotically negligible regarding the choice of
good bandwidths, and, hence, we focus on density estimation.
There are quite a number of data-driven bandwidth selection methods for
kernel density estimates. cross-validation is a popular one and frequently proposed
to estimate the optimal bandwidth. However, if the sample size is very
large, it becomes computational expensive. In material science, in particular,
it is very common to have several thousand up to several million points.
Another type of bandwidth selection is a solve-the-equation plug-in approach
which involves replacing the unknown quantities in the asymptotically optimal
bandwidth formula by their estimates.
In this thesis, we develop such an iterative fast plug-in algorithm for estimating
the optimal global and local bandwidth for density and intensity estimation with a focus on 2- and 3-dimensional data. It is based on a detailed
asymptotics of the estimators of the intensity function and of its second
derivatives and integrals of second derivatives which appear in the formulae
for asymptotically optimal bandwidths. These asymptotics are utilised to determine
the exact number of iteration steps and some tuning parameters. For
both global and local case, fewer than 10 iterations suffice. Simulation studies
show that the estimated intensity by local bandwidth can better indicate
the variation of local intensity than that by global bandwidth. Finally, the
algorithm is applied to two real data sets from test bodies of fibre-reinforced
high-performance concrete, clearly showing some inhomogeneity of the fibre
intensity.

The thesis studies change points in absolute time for censored survival data with some contributions to the more common analysis of change points with respect to survival time. We first introduce the notions and estimates of survival analysis, in particular the hazard function and censoring mechanisms. Then, we discuss change point models for survival data. In the literature, usually change points with respect to survival time are studied. Typical examples are piecewise constant and piecewise linear hazard functions. For that kind of models, we propose a new algorithm for numerical calculation of maximum likelihood estimates based on a cross entropy approach which in our simulations outperforms the common Nelder-Mead algorithm.
Our original motivation was the study of censored survival data (e.g., after diagnosis of breast cancer) over several decades. We wanted to investigate if the hazard functions differ between various time periods due, e.g., to progress in cancer treatment. This is a change point problem in the spirit of classical change point analysis. Horváth (1998) proposed a suitable change point test based on estimates of the cumulative hazard function. As an alternative, we propose similar tests based on nonparametric estimates of the hazard function. For one class of tests related to kernel probability density estimates, we develop fully the asymptotic theory for the change point tests. For the other class of estimates, which are versions of the Watson-Leadbetter estimate with censoring taken into account and which are related to the Nelson-Aalen estimate, we discuss some steps towards developing the full asymptotic theory. We close by applying the change point tests to simulated and real data, in particular to the breast cancer survival data from the SEER study.

Nonwoven materials are used as filter media which are the key component of automotive filters such as air filters, oil filters, and fuel filters. Today, the advanced engine technologies require innovative filter media with higher performances. A virtual microstructure of the nonwoven filter medium, which has similar filter properties as the existing material, can be used to design new filter media from existing media. Nonwoven materials considered in this thesis prominently feature non-overlapping fibers, curved fibers, fibers with circular cross section, fibers of apparently infinite length, and fiber bundles. To this end, as part of this thesis, we extend the Altendorf-Jeulin individual fiber model to incorporate all the above mentioned features. The resulting novel stochastic 3D fiber model can generate geometries with good visual resemblance of real filter media. Furthermore, pressure drop, which is one of the important physical properties of the filter, simulated numerically on the computed tomography (CT) data of the real nonwoven material agrees well (with a relative error of 8%) with the pressure drop simulated in the generated microstructure realizations from our model.
Generally, filter properties for the CT data and generated microstructure realizations are computed using numerical simulations. Since numerical simulations require extensive system memory and computation time, it is important to find the representative domain size of the generated microstructure for a required filter property. As part of this thesis, simulation and a statistical approach are used to estimate the representative domain size of our microstructure model. Precisely, the representative domain size with respect to the packing density, the pore size distribution, and the pressure drop are considered. It turns out that the statistical approach can be used to estimate the representative domain size for the given property more precisely and using less generated microstructures than the purely simulation based approach.
Among the various properties of fibrous filter media, fiber thickness and orientation are important characteristics which should be considered in design and quality assurance of filter media. Automatic analysis of images from scanning electron microscopy (SEM) is a suitable tool in that context. Yet, the accuracy of such image analysis tools cannot be judged based on images of real filter media since their true fiber thickness and orientation can never be known accurately. A solution is to employ synthetically generated models for evaluation. By combining our 3D fiber system model with simulation of the SEM imaging process, quantitative evaluation of the fiber thickness and orientation measurements becomes feasible. We evaluate the state-of-the-art automatic thickness and orientation estimation method that way.

Following the ideas presented in Dahlhaus (2000) and Dahlhaus and Sahm (2000) for time series, we build a Whittle-type approximation of the Gaussian likelihood for locally stationary random fields. To achieve this goal, we extend a Szegö-type formula, for the multidimensional and local stationary case and secondly we derived a set of matrix approximations using elements of the spectral theory of stochastic processes. The minimization of the Whittle likelihood leads to the so-called Whittle estimator \(\widehat{\theta}_{T}\). For the sake of simplicity we assume known mean (without loss of generality zero mean), and hence \(\widehat{\theta}_{T}\) estimates the parameter vector of the covariance matrix \(\Sigma_{\theta}\).
We investigate the asymptotic properties of the Whittle estimate, in particular uniform convergence of the likelihoods, and consistency and Gaussianity of the estimator. A main point is a detailed analysis of the asymptotic bias which is considerably more difficult for random fields than for time series. Furthemore, we prove in case of model misspecification that the minimum of our Whittle likelihood still converges, where the limit is the minimum of the Kullback-Leibler information divergence.
Finally, we evaluate the performance of the Whittle estimator through computational simulations and estimation of conditional autoregressive models, and a real data application.

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.

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.

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.