Kaiserslautern - Fachbereich Mathematik
Refine
Year of publication
Document Type
- Doctoral Thesis (292) (remove)
Has Fulltext
- yes (292)
Keywords
Faculty / Organisational entity
The thesis is concerned with multiscale approximation by means of radial basis functions on hierarchically structured spherical grids. A new approach is proposed to construct a biorthogonal system of locally supported zonal functions. By use of this biorthogonal system of locally supported zonal functions, a spherical fast wavelet transform (SFWT) is established. Finally, based on the wavelet analysis, geophysically and geodetically relevant problems involving rotation-invariant pseudodifferential operators are shown to be efficiently and economically solvable.
The desire to model in ever increasing detail geometrical and physical features has lead to a steady increase in the number of points used in field solvers. While many solvers have been ported to parallel machines, grid generators have left behind. Sequential generation of meshes of large size is extremely problematic both in terms of time and memory requirements. Therefore, the need for developing parallel mesh generation technique is well justified. In this work a novel algorithm is presented for automatic parallel generation of tetrahedral computational meshes based on geometrical domain decomposition. It has a potential to remove this bottleneck. Different domain decomposition approaches and criteria have been investigated. Questions regarding time and memory consumption, efficiency of computations and quality of generated surface and volume meshes have been considered. As a result of the work parTgen (partitioner and parallel tetrahedral mesh generator) software package based on the developed algorithm has been created. Several real-life examples of relatively complex structures involving large meshes (of order 10^7-10^8 elements) are given. It has been shown that high mesh quality is achieved. Memory and time consumption are reduced significantly, and parallel algorithm is efficient.
This dissertation is intended to transport the theory of Serre functors into the context of A-infinity-categories. We begin with an introduction to multicategories and closed multicategories, which form a framework in which the theory of A-infinity-categories is developed. We prove that (unital) A-infinity-categories constitute a closed symmetric multicategory. We define the notion of A-infinity-bimodule similarly to Tradler and show that it is equivalent to an A-infinity-functor of two arguments which takes values in the differential graded category of complexes of k-modules, where k is a commutative ground ring. Serre A-infinity-functors are defined via A-infinity-bimodules following ideas of Kontsevich and Soibelman. We prove that a unital closed under shifts A-infinity-category over a field admits a Serre A-infinity-functor if and only if its homotopy category admits an ordinary Serre functor. The proof uses categories and Serre functors enriched in the homotopy category of complexes of k-modules. Another important ingredient is an A-infinity-version of the Yoneda Lemma.
The present work deals with the (global and local) modeling of the windfield on the real topography of Rheinland-Pfalz. Thereby the focus is on the construction of a vectorial windfield from low, irregularly distributed data given on a topographical surface. The developed spline procedure works by means of vectorial (homogeneous, harmonic) polynomials (outer harmonics) which control the oscillation behaviour of the spline interpoland. In the process the characteristic of the spline curvature which defines the energy norm is assumed to be on a sphere inside the Earth interior and not on the Earth’s surface. The numerical advantage of this method arises from the maximum-minimum principle for harmonic functions.
In the theoretical part of this thesis, the difference of the solutions of the elastic and the elastoplastic boundary value problem is analysed, both for linear kinematic and combined linear kinematic and isotropic hardening material. We consider both models in their quasistatic, rate-independent formulation with linearised geometry. The main result of the thesis is, that the differences of the physical obervables (the stresses, strains and displacements) can be expressed as composition of some linear operators and play operators with respect to the exterior forces. Explicit homotopies between both solutions are presented. The main analytical devices are Lipschitz estimates for the stop and the play operator. We present some generalisations of the standard estimates. They allow different input functions, different initial memories and different scalar products. Thereby, the underlying time involving function spaces are the Sobolov spaces of first order with arbitrary integrability exponent between one and infinity. The main results can easily be generalised for the class of continuous functions with bounded total variation. In the practical part of this work, a method to correct the elastic stress tensor over a long time interval at some chosen points of the body is presented and analysed. In contrast to widespread uniaxial corrections (Neuber or ESED), our method takes multiaxiality phenomena like cyclic hardening/softening, ratchetting and non-masing behaviour into account using Jiang's model of elastoplasticity. It can be easily adapted to other constitutive elastoplastic material laws. The theory for our correction model is developped for linear kinematic hardening material, for which error estimated are derived. Our numerical algorithm is very fast and designed for the case that the elastic stress is piecewise linear. The results for the stresses can be significantly improved with Seeger's empirical strain constraint. For the improved model, a simple predictor-correcor algorithm for smooth input loading is established.
The main aim of this work was to obtain an approximate solution of the seismic traveltime tomography problems with the help of splines based on reproducing kernel Sobolev spaces. In order to be able to apply the spline approximation concept to surface wave as well as to body wave tomography problems, the spherical spline approximation concept was extended for the case where the domain of the function to be approximated is an arbitrary compact set in R^n and a finite number of discontinuity points is allowed. We present applications of such spline method to seismic surface wave as well as body wave tomography, and discuss the theoretical and numerical aspects of such applications. Moreover, we run numerous numerical tests that justify the theoretical considerations.
Zwei zentrale Probleme der modernen Finanzmathematik sind die Portfolio-Optimierung und die Optionsbewertung. Während es bei der Portfolio-Optimierung darum geht, das Vermögen optimal auf verschiedene Anlagemöglichkeiten zu verteilen, versucht die Optionsbewertung faire Preise von derivativen Finanzinstrumenten zu bestimmen. In dieser Arbeit werden Fragestellungen aus beiden dieser Themenbereiche bearbeitet. Die Arbeit beginnt mit einem Kapitel über Grundlagen, in dem zum Beispiel das Portfolio-Problem von Merton dargestellt und die Black/Scholes-Formel zur Optionsbewertung hergeleitet wird. In Kapitel 2 wird das Portfolio-Problem von Morton und Pliska betrachtet, die in das Merton-Modell fixe Transaktionskosten eingeführt haben. Dabei muß der Investor bei jeder Transaktion einen fixen Anteil vom derzeitigen Vermögen als Kosten abführen. Es wird die asymptotische Approximation dieses Modells von Atkinson und Wilmott vorgestellt und die optimale Portfoliostrategie aus den Marktparametern hergeleitet. Danach werden die tatsächlichen Transaktionskosten abgeschätzt und ein User Guide zur praktischen Anwendung dieses Transaktionskostenmodells angegeben. Zum Schluß wird das Modell numerisch analysiert, indem unter anderem die erwartete Handelszeit und die Güte der Abschätzung der tatsächlichen Transaktionskosten berechnet werden. Ein Portfolio-Problem mit internationalen Märkten wird in Kapitel 3 vorgestellt. Dem Investor steht zusätzlich zu seinem Heimatland noch ein weiteres Land für seine Vermögensanlagen zur Verfügung. Dabei werden die Preisprozesse für die ausländischen Wertpapiere mit einem stochastischen Wechselkurs in die Heimatwährung umgerechnet. In einer statischen Analyse wird unter anderem berechnet, wieviel weniger Vermögen der Investor benötigt, um das gleiche erwartete Endvermögen zu erhalten wie in dem Fall, wenn ihm keine Auslandsanlagen zur Verfügung stehen. Kapitel 4 behandelt drei verschiedene Portfolio-Probleme mit Sprung-Diffusions-Prozessen. Nach der Herleitung eines Verifikationssatzes wird das Problem bei Anlagemöglichkeit in eine Aktie und in ein Geldmarktkonto jeweils für eine konstante und eine stochastische Zinsrate untersucht. Im ersten Fall wird eine implizite Darstellung für den optimalen Portfolioprozeß und eine Bedingung angegeben, unter der diese Darstellung eindeutig lösbar ist. Außerdem wird der optimale Portfolioprozeß für verschiedene Verteilungen für die Sprunghöhe untersucht. Im Falle einer stochastischen Zinsrate kann nur ein Kandidat für den optimalen Lösungsprozeß angeben werden. Dieser hat wieder eine implizite Darstellung. Das letzte Portfolio-Problem ist eine Abwandlung des Modells aus Kapitel 3. Wird dort der Wechselkurs durch eine geometrisch Brownsche Bewegung modelliert, ist er hier ein reiner Sprungprozeß. Es wird wieder der optimale Portfolioprozeß hergeleitet, wobei ein Anteil davon unter Umständen nur numerisch lösbar ist. Eine hinreichende Bedingung für die Lösbarkeit wird angegeben. In Kapitel 5 werden verschiedene Bewertungsansätze für Optionen auf Bondindizes präsentiert. Es wird eine Methode vorgestellt, mit der die Optionen anhand von Marktpreisen bewertet werden können. Für den Fall, daß es nicht genug Marktpreise gibt, wird ein Verfahren angegeben, um den Bondindex realitätsnah zu simulieren und künstliche Marktpreise zu erzeugen. Diese Preise können dann für eine Kalibrierung verwendet werden.
Diese Arbeit beschäftigt sich mit Methoden zur Klassifikation von Ovoiden in quadratischen Räumen. Die Anwendung der dazu entwickelten Algorithmen erfolgt hauptsächlich in achtdimensionalen Räumen speziell über den Körpern GF(7), GF(8) und GF(9). Zu verschiedenen, zumeist kleinen, zyklischen Gruppen werden hier die unter diesen Gruppen invarianten Ovoide bestimmt. Die bei dieser Suche auftretenden Ovoide sind alle bereits bekannt. Es ergeben sich jedoch Restriktionen an die Stabilisatoren gegebenenfalls existierender, unbekannter Ovoide.
In this thesis, the quasi-static Biot poroelasticity system in bounded multilayered domains in one and three dimensions is studied. In more detail, in the one-dimensional case, a finite volume discretization for the Biot system with discontinuous coefficients is derived. The discretization results in a difference scheme with harmonic averaging of the coefficients. Detailed theoretical analysis of the obtained discrete model is performed. Error estimates, which establish convergence rates for both primary as well as flux unknowns are derived. Besides, modified and more accurate discretizations, which can be applied when the interface position coincides with a grid node, are obtained. These discretizations yield second order convergence of the fluxes of the problem. Finally, the solver for the solution of the produced system of linear equations is developed and extensively tested. A number of numerical experiments, which confirm the theoretical considerations are performed. In the three-dimensional case, the finite volume discretization of the system involves construction of special interpolating polynomials in the dual volumes. These polynomials are derived so that they satisfy the same continuity conditions across the interface, as the original system of PDEs. This technique allows to obtain such a difference scheme, which provides accurate computation of the primary as well as of the flux unknowns, including the points adjacent to the interface. Numerical experiments, based on the obtained discretization, show second order convergence for auxiliary problems with known analytical solutions. A multigrid solver, which incorporates the features of the discrete model, is developed in order to solve efficiently the linear system, produced by the finite volume discretization of the three-dimensional problem. The crucial point is to derive problem-dependent restriction and prolongation operators. Such operators are a well-known remedy for the scalar PDEs with discontinuous coefficients. Here, these operators are derived for the system of PDEs, taking into account interdependence of different unknowns within the system. In the derivation, the interpolating polynomials from the finite volume discretization are employed again, linking thus the discretization and the solution processes. The developed multigrid solver is tested on several model problems. Numerical experiments show that, due to the proper problem-dependent intergrid transfer, the multigrid solver is robust with respect to the discontinuities of the coefficients of the system. In the end, the poroelasticity system with discontinuous coefficients is used to model a real problem. The Biot model, describing this problem, is treated numerically, i.e., discretized by the developed finite volume techniques and then solved by the constructed multigrid solver. Physical characteristics of the process, such as displacement of the skeleton, pressure of the fluid, components of the stress tensor, are calculated and then presented at certain cross-sections.
In the thesis the author presents a mathematical model which describes the behaviour of the acoustical pressure (sound), produced by a bass loudspeaker. The underlying physical propagation of sound is described by the non--linear isentropic Euler system in a Lagrangian description. This system is expanded via asymptotical analysis up to third order in the displacement of the membrane of the loudspeaker. The differential equations which describe the behaviour of the key note and the first order harmonic are compared to classical results. The boundary conditions, which are derived up to third order, are based on the principle that the small control volume sticks to the boundary and is allowed to move only along it. Using classical results of the theory of elliptic partial differential equations, the author shows that under appropriate conditions on the input data the appropriate mathematical problems admit, by the Fredholm alternative, unique solutions. Moreover, certain regularity results are shown. Further, a novel Wave Based Method is applied to solve appropriate mathematical problems. However, the known theory of the Wave Based Method, which can be found in the literature, so far, allowed to apply WBM only in the cases of convex domains. The author finds the criterion which allows to apply the WBM in the cases of non--convex domains. In the case of 2D problems we represent this criterion as a small proposition. With the aid of this proposition one is able to subdivide arbitrary 2D domains such that the number of subdomains is minimal, WBM may be applied in each subdomain and the geometry is not altered, e.g. via polygonal approximation. Further, the same principles are used in the case of 3D problem. However, the formulation of a similar proposition in cases of 3D problems has still to be done. Next, we show a simple procedure to solve an inhomogeneous Helmholtz equation using WBM. This procedure, however, is rather computationally expensive and can probably be improved. Several examples are also presented. We present the possibility to apply the Wave Based Technique to solve steady--state acoustic problems in the case of an unbounded 3D domain. The main principle of the classical WBM is extended to the case of an external domain. Two numerical examples are also presented. In order to apply the WBM to our problems we subdivide the computational domain into three subdomains. Therefore, on the interfaces certain coupling conditions are defined. The description of the optimization procedure, based on the principles of the shape gradient method and level set method, and the results of the optimization finalize the thesis.
In this dissertation we present analysis of macroscopic models for slow dense granular flow. Models are derived from plasticity theory with yield condition and flow rule. Corner stone equations are conservation of mass and conservation of momentum with special constitutive law. Such models are considered in the class of generalised Newtonian fluids, where viscosity depends on the pressure and modulo of the strain-rate tensor. We showed the hyperbolic nature for the evolutionary model in 1D and ill-posed behaviour for 2D and 3D. The steady state equations are always hyperbolic. In the 2D problem we derived a prototype nonlinear backward parabolic equation for the velocity and the similar equation for the shear-rate. Analysis of derived PDE showed the finite blow up time. Blow up time depends on the initial condition. Full 2D and antiplane 3D model were investigated numerically with finite element method. For 2D model we showed the presence of boundary layers. Antiplane 3D model was investigated with the Runge Kutta Discontinuous Galerkin method with mesh addoption. Numerical results confirmed that such a numerical method can be a good choice for the simulations of the slow dense granular flow.
The lattice Boltzmann method (LBM) is a numerical solver for the Navier-Stokes equations, based on an underlying molecular dynamic model. Recently, it has been extended towardsthe simulation of complex fluids. We use the asymptotic expansion technique to investigate the standard scheme, the initialization problem and possible developments towards moving boundary and fluid-structure interaction problems. At the same time, it will be shown how the mathematical analysis can be used to understand and improve the algorithm. First of all, we elaborate the tool "asymptotic analysis", proposing a general formulation of the technique and explaining the methods and the strategy we use for the investigation. A first standard application to the LBM is described, which leads to the approximation of the Navier-Stokes solution starting from the lattice Boltzmann equation. As next, we extend the analysis to investigate origin and dynamics of initial layers. A class of initialization algorithms to generate accurate initial values within the LB framework is described in detail. Starting from existing routines, we will be able to improve the schemes in term of efficiency and accuracy. Then we study the features of a simple moving boundary LBM. In particular, we concentrate on the initialization of new fluid nodes created by the variations of the computational fluid domain. An overview of existing possible choices is presented. Performing a careful analysis of the problem we propose a modified algorithm, which produces satisfactory results. Finally, to set up an LBM for fluid structure interaction, efficient routines to evaluate forces are required. We describe the Momentum Exchange algorithm (MEA). Precise accuracy estimates are derived, and the analysis leads to the construction of an improved method to evaluate the interface stresses. In conclusion, we test the defined code and validate the results of the analysis on several simple benchmarks. From the theoretical point of view, in the thesis we have developed a general formulation of the asymptotic expansion, which is expected to offer a more flexible tool in the investigation of numerical methods. The main practical contribution offered by this work is the detailed analysis of the numerical method. It allows to understand and improve the algorithms, and construct new routines, which can be considered as starting points for future researches.
This thesis introduces so-called cone scalarising functions. They are by construction compatible with a partial order for the outcome space given by a cone. The quality of the parametrisations of the efficient set given by the cone scalarising functions are then investigated. Here, the focus lies on the (weak) efficiency of the generated solutions, the reachability of effiecient points and continuity of the solution set. Based on cone scalarising functions Pareto Navigation a novel, interactive, multiobjective optimisation method is proposed. It changes the ordering cone to realise bounds on partial tradeoffs. Besides, its use of an equality constraint for the changing component of the reference point is a new feature. The efficiency of its solutions, the reachability of efficient solutions and continuity is then analysed. Potential problems are demonstrated using a critical example. Furthermore, the use of Pareto Navigation in a two-phase approach and for nonconvex problems is discussed. Finally, its application for intensity-modulated radiotherapy planning is described. Thereby, its realisation in a graphical user interface is shown.
Die Arbeit beschäftigt sich mit den Charakteren des Normalisators und des Zentralisators eines Sylowtorus. Dabei wird jede Gruppe G vom Lie-Typ als Fixpunktgruppe einer einfach-zusammenhängenden einfachen Gruppe unter einer Frobeniusabbildung aufgefaßt. Für jeden Sylowtorus S der algebraischen Gruppe wird gezeigt, dass die irreduziblen Charaktere des Zentralisators von S in G sich auf ihre Trägheitsgruppe im Normalisator von S fortsetzen. Diese Fragestellung entsteht aus dem Studium der Höhe 0 Charaktere bei endlichen reduktiven Gruppen vom Lie-Typ im Zusammenhang mit der McKay-Vermutung. Neuere Resultate von Isaacs, Malle und Navarro führen diese Vermutung auf eine Eigenschaft von einfachen Gruppen zurück, die sie dann für eine Primzahl gut nennen. Bei Gruppen vom Lie-Typ zeigt das obige Resultat zusammen mit einer aktuellen Arbeit von Malle einige dabei wichtige und notwendige Eigenschaften. Anhand der Steinberg-Präsentation werden vor allem bei den klassischen Gruppen genauere Aussagen über die Struktur des Zentralisators und des Normalisators eines Sylowtorus bewiesen. Wichtig dabei ist die von Tits eingeführte erweiterte Weylgruppe, die starke Verbindungen zu Zopfgruppen besitzt. Das Resultat wird in zahlreichen Einzelfallbetrachtungen gezeigt, bei denen in dieser Arbeit bewiesene Vererbungsregeln von Fortsetzbarkeitseigenschaften benutzt werden.
The topic of this thesis is the coupling of an atomistic and a coarse scale region in molecular dynamics simulations with the focus on the reflection of waves at the interface between the two scales and the velocity of waves in the coarse scale region for a non-equilibrium process. First, two models from the literature for such a coupling, the concurrent coupling of length scales and the bridging scales method are investigated for a one dimensional system with harmonic interaction. It turns out that the concurrent coupling of length scales method leads to the reflection of fine scale waves at the interface, while the bridging scales method gives an approximated system that is not energy conserving. The velocity of waves in the coarse scale region is in both models not correct. To circumvent this problems, we present a coupling based on the displacement splitting of the bridging scales method together with choosing appropriate variables in orthogonal subspaces. This coupling allows the derivation of evolution equations of fine and coarse scale degrees of freedom together with a reflectionless boundary condition at the interface directly from the Lagrangian of the system. This leads to an energy conserving approximated system with a clear separation between modeling errors an errors due to the numerical solution. Possible approximations in the Lagrangian and the numerical computation of the memory integral and other numerical errors are discussed. We further present a method to choose the interpolation from coarse to atomistic scale in such a way, that the fine scale degrees of freedom in the coarse scale region can be neglected. The interpolation weights are computed by comparing the dispersion relations of the coarse scale equations and the fully atomistic system. With this new interpolation weights, the number of degrees of freedom can be drastically reduced without creating an error in the velocity of the waves in the coarse scale region. We give an alternative derivation of the new coupling with the Mori-Zwanzig projection operator formalism, and explain how the method can be extended to non-zero temperature simulations. For the comparison of the results of the approximated with the fully atomistic system, we use a local stress tensor and the energy in the atomistic region. Examples for the numerical solution of the approximated system for harmonic potentials are given in one and two dimensions.
This thesis contains the mathematical treatment of a special class of analog microelectronic circuits called translinear circuits. The goal is to provide foundations of a new coherent synthesis approach for this class of circuits. The mathematical methods of the suggested synthesis approach come from graph theory, combinatorics, and from algebraic geometry, in particular symbolic methods from computer algebra. Translinear circuits form a very special class of analog circuits, because they rely on nonlinear device models, but still allow a very structured approach to network analysis and synthesis. Thus, translinear circuits play the role of a bridge between the "unknown space" of nonlinear circuit theory and the very well exploited domain of linear circuit theory. The nonlinear equations describing the behavior of translinear circuits possess a strong algebraic structure that is nonetheless flexible enough for a wide range of nonlinear functionality. Furthermore, translinear circuits offer several technical advantages like high functional density, low supply voltage and insensitivity to temperature. This unique profile is the reason that several authors consider translinear networks as the key to systematic synthesis methods for nonlinear circuits. The thesis proposes the usage of a computer-generated catalog of translinear network topologies as a synthesis tool. The idea to compile such a catalog has grown from the observation that on the one hand, the topology of a translinear network must satisfy strong constraints which severely limit the number of "admissible" topologies, in particular for networks with few transistors, and on the other hand, the topology of a translinear network already fixes its essential behavior, at least for static networks, because the so-called translinear principle requires the continuous parameters of all transistors to be the same. Even though the admissible topologies are heavily restricted, it is a highly nontrivial task to compile such a catalog. Combinatorial techniques have been adapted to undertake this task. In a catalog of translinear network topologies, prototype network equations can be stored along with each topology. When a circuit with a specified behavior is to be designed, one can search the catalog for a network whose equations can be matched with the desired behavior. In this context, two algebraic problems arise: To set up a meaningful equation for a network in the catalog, an elimination of variables must be performed, and to test whether a prototype equation from the catalog and a specified equation of desired behavior can be "matched", a complex system of polynomial equations must be solved, where the solutions are restricted to a finite set of integers. Sophisticated algorithms from computer algebra are applied in both cases to perform the symbolic computations. All mentioned algorithms have been implemented using C++, Singular, and Mathematica, and are successfully applied to actual design problems of humidity sensor circuitry at Analog Microelectronics GmbH, Mainz. As result of the research conducted, an exhaustive catalog of all static formal translinear networks with at most eight transistors is available. The application for the humidity sensor system proves the applicability of the developed synthesis approach. The details and implementations of the algorithms are worked out only for static networks, but can easily be adopted for dynamic networks as well. While the implementation of the combinatorial algorithms is stand-alone software written "from scratch" in C++, the implementation of the algebraic algorithms, namely the symbolic treatment of the network equations and the match finding, heavily rely on the sophisticated Gröbner basis engine of Singular and thus on more than a decade of experience contained in a special-purpose computer algebra system. It should be pointed out that the thesis contains the new observation that the translinear loop equations of a translinear network are precisely represented by the toric ideal of the network's translinear digraph. Altogether, this thesis confirms and strengthenes the key role of translinear circuits as systematically designable nonlinear circuits.
The fast development of the financial markets in the last decade has lead to the creation of a variety of innovative interest rate related products that require advanced numerical pricing methods. Examples in this respect are products with a complicated strong path-dependence such as a Target Redemption Note, a Ratchet Cap, a Ladder Swap and others. On the other side, the usage of the standard in the literature one-factor Hull and White (1990) type of short rate models allows only for a perfect correlation between all continuously compounded spot rates or Libor rates and thus are not suited for pricing innovative products depending on several Libor rates such as for example a "steepener" option. One possible solution to this problem deliver the two-factor short rate models and in this thesis we consider a two-factor Hull and White (1990) type of a short rate process derived from the Heath, Jarrow, Morton (1992) framework by limiting the volatility structure of the forward rate process to a deterministic one. In this thesis, we often choose to use a variety of modified (binomial, trinomial and quadrinomial) tree constructions as a main numerical pricing tool due to their flexibility and fast convergence and (when there is no closed-form solution) compare their results with fine grid Monte Carlo simulations. For the purpose of pricing the already mentioned innovative short-rate related products, in this thesis we offer and examine two different lattice construction methods for the two-factor Hull-White type of a short rate process which are able to deal easily both with modeling of the mean-reversion of the underlying process and with the strong path-dependence of the priced options. Additionally, we prove that the so-called rotated lattice construction method overcomes the typical for the existing two-factor tree constructions problem with obtaining negative "risk-neutral probabilities". With a variety of numerical examples, we show that this leads to a stability in the results especially in cases of high volatility parameters and negative correlation between the base factors (which is typically the case in reality). Further, noticing that Chan et al (1992) and Ritchken and Sankarasubramanian (1995) showed that option prices are sensitive to the level of the short rate volatility, we examine the pricing of European and American options where the short rate process has a volatility structure of a Cheyette (1994) type. In this relation, we examine the application of the two offered lattice construction methods and compare their results with the Monte Carlo simulation ones for a variety of examples. Additionally, for the pricing of American options with the Monte Carlo method we expand and implement the simulation algorithm of Longstaff and Schwartz (2000). With a variety of numerical examples we compare again the stability and the convergence of the different lattice construction methods. Dealing with the problems of pricing strongly path-dependent options, we come across the cumulative Parisian barrier option pricing problem. We notice that in their classical form, the cumulative Parisian barrier options have been priced both analytically (in a quasi closed form) and with a tree approximation (based on the Forward Shooting Grid algorithm, see e.g. Hull and White (1993), Kwok and Lau (2001) and others). However, we offer an additional tree construction method which can be seen as a direct binomial tree integration that uses the analytically calculated conditional survival probabilities. The advantage of the offered method is on one side that the conditional survival probabilities are easier to calculate than the closed-form solution itself and on the other side that this tree construction is very flexible in the sense that it allows easy incorporation of additional features such as e.g a forward starting one. The obtained results are better than the Forward Shooting Grid tree ones and are very close to the analytical quasi closed form solution. Finally, we pay our attention to pricing another type of innovative interest rate alike products - namely the Longevity bond - whose coupon payments depend on the survival function of a given cohort. Due to the lack of a market for mortality, for the pricing of the Longevity bonds we develop (following Korn, Natcheva and Zipperer (2006)) a framework that contains principles from both Insurance and Financial mathematic. Further on, we calibrate the existing models for the stochastic mortality dynamics to historical German data and additionally offer new stochastic extensions of the classical (deterministic) models of mortality such as the Gompertz and the Makeham one. Finally, we compare and analyze the results of the application of all considered models to the pricing of a Longevity bond on the longevity of the German males.
In this thesis, we have dealt with two modeling approaches of the credit risk, namely the structural (firm value) and the reduced form. In the former one, the firm value is modeled by a stochastic process and the first hitting time of this stochastic process to a given boundary defines the default time of the firm. In the existing literature, the stochastic process, triggering the firm value, has been generally chosen as a diffusion process. Therefore, on one hand it is possible to obtain closed form solutions for the pricing problems of credit derivatives and on the other hand the optimal capital structure of a firm can be analysed by obtaining closed form solutions of firm's corporate securities such as; equity value, debt value and total firm value, see Leland(1994). We have extended this approach by modeling the firm value as a jump-diffusion process. The choice of the jump-diffusion process was a crucial step to obtain closed form solutions for corporate securities. As a result, we have chosen a jump-diffusion process with double exponentially distributed jump heights, which enabled us to analyse the effects of jump on the optimal capital structure of a firm. In the second part of the thesis, by following the reduced form models, we have assumed that the default is triggered by the first jump of a Cox process. Further, by following Schönbucher(2005), we have modeled the forward default intensity of a firm as a geometric Brownian motion and derived pricing formulas for credit default swap options in a more general setup than the ones in Schönbucher(2005).
In this thesis diverse problems concerning inflation-linked products are dealt with. To start with, two models for inflation are presented, including a geometric Brownian motion for consumer price index itself and an extended Vasicek model for inflation rate. For both suggested models the pricing formulas of inflation-linked products are derived using the risk-neutral valuation techniques. As a result Black and Scholes type closed form solutions for a call option on inflation index for a Brownian motion model and inflation evolution for an extended Vasicek model as well as for an inflation-linked bond are calculated. These results have been already presented in Korn and Kruse (2004) [17]. In addition to these inflation-linked products, for the both inflation models the pricing formulas of a European put option on inflation, an inflation cap and floor, an inflation swap and an inflation swaption are derived. Consequently, basing on the derived pricing formulas and assuming the geometric Brownian motion process for an inflation index, different continuous-time portfolio problems as well as hedging problems are studied using the martingale techniques as well as stochastic optimal control methods. These utility optimization problems are continuous-time portfolio problems in different financial market setups and in addition with a positive lower bound constraint on the final wealth of the investor. When one summarizes all the optimization problems studied in this work, one will have the complete picture of the inflation-linked market and both counterparts of market-participants, sellers as well as buyers of inflation-linked financial products. One of the interesting results worth mentioning here is naturally the fact that a regular risk-averse investor would like to sell and not buy inflation-linked products due to the high price of inflation-linked bonds for example and an underperformance of inflation-linked bonds compared to the conventional risk-free bonds. The relevance of this observation is proved by investigating a simple optimization problem for the extended Vasicek process, where as a result we still have an underperforming inflation-linked bond compared to the conventional bond. This situation does not change, when one switches to an optimization of expected utility from the purchasing power, because in its nature it is only a change of measure, where we have a different deflator. The negativity of the optimal portfolio process for a normal investor is in itself an interesting aspect, but it does not affect the optimality of handling inflation-linked products compared to the situation not including these products into investment portfolio. In the following, hedging problems are considered as a modeling of the other half of inflation market that is inflation-linked products buyers. Natural buyers of these inflation-linked products are obviously institutions that have payment obligations in the future that are inflation connected. That is why we consider problems of hedging inflation-indexed payment obligations with different financial assets. The role of inflation-linked products in the hedging portfolio is shown to be very important by analyzing two alternative optimal hedging strategies, where in the first one an investor is allowed to trade as inflation-linked bond and in the second one he is not allowed to include an inflation-linked bond into his hedging portfolio. Technically this is done by restricting our original financial market, which is made of a conventional bond, inflation index and a stock correlated with inflation index, to the one, where an inflation index is excluded. As a whole, this thesis presents a wide view on inflation-linked products: inflation modeling, pricing aspects of inflation-linked products, various continuous-time portfolio problems with inflation-linked products as well as hedging of inflation-related payment obligations.
This thesis discusses methods for the classification of finite projective planes via exhaustive search. In the main part the author classifies all projective planes of order 16 admitting a large quasiregular group of collineations. This is done by a complete search using the computer algebra system GAP. Computational methods for the construction of relative difference sets are discussed. These methods are implemented in a GAP-package, which is available separately. As another result --found in cooperation with U. Dempwolff-- the projective planes defined by planar monomials are classified. Furthermore the full automorphism group of the non-translation planes defined by planar monomials are classified.