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This thesis deals with modeling aspects of generalized Newtonian and of non-Newtonian fluids, as well as with development and validation of algorithms used in simulation of such fluids. The main contribution in the modeling part are the introduction and analysis of a new model for the generalized Newtonian fluids, where constitutive equation is of an algebraic form. Distinction between shear and extensional viscosities leads to anisotropic viscosity model. It can be considered as a natural extension of the well known (isotropic viscosity) Carreau model, which deals only with shear viscosity properties of the fluid. The proposed model takes additionally into account extensional viscosity properties. Numerical results show that the anisotropic viscosity model gives much better agreement with experimental observations than the isotropic one. Another contribution of the thesis consists of the development and analysis of robust and reliable algorithms for simulation of generalized Newtonian fluids. For such fluids the momentum equations are strongly coupled through mixed derivatives appearing in the viscous term (unlike the case of Newtonian fluids). It is shown in this thesis, that a careful treatment of those derivatives is essential in deriving robust algorithms. A modification of a standard SIMPLE-like algorithm is given, where all the viscous terms from the momentum equations are discretized in an implicit manner. Moreover, it is shown that a block diagonal preconditioner to the viscous operator is good enough to be used in simulations. Furthermore, different solution techniques, namely projection type methods (consists of solving momentum equations and pressure correction equation) and fully coupled methods (momentum and continuity equations are solved together), are compared. It is shown, that explicit discretization of the mixed derivatives lead to stability problems. Further, analytical estimates of eigenvalue distribution for three different preconditioners, applied to the transformed system arising after discretization and linearization of the momentum and continuity equations, are provided. We propose to apply a block Gauss-Seidel preconditioner to the transformed system. The analysis shows, that this preconditioner is able to cluster eigenvalues around unity independent of the transformation step. It is not the case for other preconditioners applied to the transformed system as discussed in the thesis. The block Gauss-Seidel preconditioner has also shown the best behavior (among all preconditioners discussed in the thesis) in numerical experiments. Further contribution consists of comparison and validation of numerical algorithms applied in simulations of non-Newtonian fluids modeled by time integral constitutive equations. Numerical results from simulations of dilute polymer solutions, described by the integral Oldroyd B model, have shown very good quantitative agreement with the results obtained by differential Oldroyd B counterpart in 4:1 planar contraction domain at low Weissenberg numbers. In this case, the Weissenberg number is changed by changing the relaxation time. However, contrary to the differential Oldroyd B model, the integral one allows to perform stable simulations also in the range of high Weissenberg numbers. Moreover, very good agreement with experimental observations has been achieved. Simulations of concentrated polymer solutions (polystyrene and polybutadiene solutions), modeled by the integral Doi Edwards model, supplemented by chain length fluctuations, have shown very good qualitative agreement with the results obtained by its differential approximation in 4:1:4 constriction domain. Again, much higher Weissenberg numbers can be achieved when the integral model is used. Moreover, very good quantitative results with experimental data of polystyrene solution for the first normal stress difference and shear viscosity defined here as the quotient of a shear stress and a shear rate. Finally, comparison of the two methods used for approximating the time integral constitutive equation, namely Deformation Field Method (DFM) and Backward Lagrangian Particle Method (BLPM), is performed. In BLPM the particle paths are recalculated at every time step of the simulations, what has never been tried before. The results have shown, that in the considered geometries both methods give similar results.

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.

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.

Discontinuities can appear in different fields of mechanics. Some examples where discontinuities arise are more obvious such as the formation of cracks. Other sources of discontinuities are less apparent such as interfaces between different materials. Furthermore continuous fields with steep gradients can also be considered as discontinuous fields. This work aims at the inclusion of arbitrary discontinuities within the finite element method. Although the finite element method is the most sophisticated numerical tool in modern engineering, the inclusion of discontinuities is still a challenging task. Traditionally within finite the framework of FE methods discontinuities are modeled explicitely by the construction of the mesh. Thus, when a fixed mesh is used, the position of the discontinuity is prescribed by the location of interelement boundaries and not by the physical situation. The simulation of crack growth requires a frequent adaption of the mesh and that can be a difficult and computationally expensive task. Thus a more flexible numerical approach is needed which leads to the mesh-independent representation of the discontinuity. A challenging field where the accurate description of discontinuities is of vital importance is the modeling of failure in engineering materials. The load capacity of a structure is limited by the material strength. If the load limit is exceeded failure zones arise and increase. Representative examples of failure mechanisms are are cracks in brittle materials or shear bands in metals or soils. Failure processes are often accompanied by a strain softening material behaviour (decreasing load carrying capacity with increasing strain at a material point). It is known that the inclusion of strain softening material behaviour within a continuum description requires regularization techniques to preserve the well- posedness of the governing equations. One possibility is the consideration of non-local or gradient terms in the constitutive equations but these approaches require a sufficiently fine discretization in the localization zone, which leads to a high numerical effort. If the extent of the failure zone and the failure process to the point of the development of discrete cracks is considered it seems reasonable to include strong discontinuities. In the framework of fracture mechanics the inclusion of displacement jumps is intuitively comprehensible. However, the modeling of localized failure processes demands the consideration of inelastic material behaviour. Cohesive zone models represent an approach which is especially suited for the incorporation within the finite element framework. It is supposed that cohesive tractions are transmitted between the discontinuity surfaces. These tractions are constitutively prescribed by a phenomenological traction separation law and thus allow for the modeling of different inelastic mechanisms, like micro-crack evolution, initiation of voids, plastic flow or crack bridging. The incorporation of a displacement discontinuity in combination with a cohesive traction separation law leads to a sound model to describe failure processes and crack propagation. Another area where the existence of discontinuities is not as obvious is the occurence of material interfaces, inclusions or holes. The accurate modeling of such internal interfaces is important to predict the mechanical behaviour of components. The present discontinuity is of different nature: the displacement field is continuous but there is a jump in the strains, which is denoted by the expression weak discontinuity. Usually in FE methods material interfaces are taken into account by the mesh construction. But if the structure exhibits multiple inclusions of complex geometry it can be advantageous if the interface does not have to be meshed. And when we look at at problems where the interface moves with time, e. g. phase transformation, the mesh-independent modeling of the weak discontinuities naturally holds major advantages. The greatest challenge in the modeling of discontinuities is their incorporation into numerical methods. The focus of the present work is the development, analysis and application of a finite element approach to model mesh-independent discontinuities. The method shall be robust and flexible to be applicable to both, strong and weak discontinuities.

Tropical geometry is a rather new field of algebraic geometry. The main idea is to replace algebraic varieties by certain piece-wise linear objects in R^n, which can be studied with the aid of combinatorics. There is hope that many algebraically difficult operations become easier in the tropical setting, as the structure of the objects seems to be simpler. In particular, tropical geometry shows promise for application in enumerative geometry. Enumerative geometry deals with the counting of geometric objects that are determined by certain incidence conditions. Until around 1990, not many enumerative questions had been answered and there was not much prospect of solving more. But then Kontsevich introduced the moduli space of stable maps which turned out to be a very useful concept for the study of enumerative geometry. A well-known problem of enumerative geometry is to determine the numbers N_cplx(d,g) of complex genus g plane curves of degree d passing through 3d+g-1 points in general position. Mikhalkin has defined the analogous number N_trop(d,g) for tropical curves and shown that these two numbers coincide (Mikhalkin's Correspondence Theorem). Tropical geometry supplies many new ideas and concepts that could be helpful to answer enumerative problems. However, as a rather new field, tropical geometry has to be studied more thoroughly. This thesis is concerned with the ``translation'' of well-known facts of enumerative geometry to tropical geometry. More precisely, the main results of this thesis are: - a tropical proof of the invariance of N_trop(d,g) of the position of the 3d+g-1 points, - a tropical proof for Kontsevich's recursive formula to compute N_trop(d,0) and - a tropical proof of Caporaso's and Harris' algorithm to compute N_trop(d,g). All results were derived in joint work with my advisor Andreas Gathmann. (Note that tropical research is not restricted to the translation of classically well-known facts, there are actually new results shown by means of tropical geometry that have not been known before. For example, Mikhalkin gave a tropical algorithm to compute the Welschinger invariant for real curves. This shows that tropical geometry can indeed be a tool for a better understanding of classical geometry.)

The primary object of this work is the development of a robust, accurate and efficient time integrator for the dynamics of flexible multibody systems. Particularly a unified framework for the computational dynamics of multibody systems consisting of mass points, rigid bodies and flexible beams forming open kinematic chains or closed loop systems is developed. In addition, it aims at the presentation of (i) a focused survey of the Lagrangian and Hamiltonian formalism for dynamics, (ii) five different methods to enforce constraints with their respective relations, and (iii) three alternative ways for the temporal discretisation of the evolution equations. The relations between the different methods for the constraint enforcement in conjunction with one specific energy-momentum conserving temporal discretisation method are proved and their numerical performances are compared by means of theoretical considerations as well as with the help of numerical examples.

The study provides insights into the dynamic processes of vascular epiphyte vegetation in two host tree species of lowland forest in Panama. Further, a novel approach is presented to examine the possible role of host tree identity in the structuring of vascular epiphyte communities: For three locally common host tree species (Socratea exorrhiza, Marila laxiflora, Perebea xanthochyma) we created null models of the expected epiphyte assemblages assuming that epiphyte colonization reflected random distribution of epiphytes in the forest. In all three tree species, abundances of the majority of epiphyte species (69 – 81 %) were indistinguishable from random, while the remaining species were about equally over- or underrepresented compared to their occurrence in the entire forest plot. Permutations based on the number of colonized trees (reflecting observed spatial patchiness) yielded similar results. Finally, a Canonical Correspondence Analysis also confirmed host-specific differences in epiphyte assemblages. In spite of pronounced preferences of some epiphytes for particular host trees, no epiphyte species was restricted to a single host. We conclude that the epiphytes on a given tree species are not simply a random sample of the local species pool, but there are no indications of host specificity either. To determine the qualitative and quantitative long-term changes in the vascular epiphyte assemblage of the host tree Socratea exorrhiza, in the lowland forest of the San Lorenzo Crane Plot, we followed the fate of the vascular epiphyte assemblage on 99 individuals of this palm species, in three censuses over the course of five years. The composition of the epiphyte assemblage changed little during the course of the study. While the similarity of epiphyte vegetation decreased on single palm individuals through time, the similarity analyzed over all palms increased. Even well-established epiphyte individuals experienced high mortality with only 46 % of the originally mapped individuals surviving the following five years. We found a positive correlation between host tree size and epiphyte richness and detected higher colonization rates of epiphytes per surface area on larger trees. Epiphyte assemblages on single Socratea exorrhiza trees were highly dynamic while the overall composition of the epiphyte vegetation on the host tree species in the study plot was rather stable. We suggest that higher recruitment rates due to localized seed dispersal by already established epiphytes on larger palms promote the colonization of epiphytes on larger palms. Given the known growth rates and mortality rates of the host tree species, the maximum time available for colonization and reproduction of epiphytes on a given Socratea exorrhiza tree is estimated to be about 60 years. Changes in the epiphyte vegetation of c. 1000 individuals of the host tree species Annona glabra at Barro Colorado Island over the course of eight year were documented by means of repeated censuses. Considerable increase in the abundance of the dominating epiphyte species and ongoing colonization of the host tree species suggests that the epiphyte vegetation has not reached a steady state in the maximal 80 years since the establishment of the host tree. Epiphyte species composition as a whole was rather stable. We disentangled the relationship between epiphyte colonization and tree size/available time for colonization with the finding that tree size explained only a low proportion of colonization while other factors like connectivity to dispersal source and time explain may explain a larger part. Epiphyte populations are patchily distributed and examined species exhibit properties of a metapopulation with asynchronous local population growth, high local population turnover, a positive relationship between regional occurrence and patch population size, and negatively correlated relationship between extinction and patch occupancy. The documented metapopulation processes highlight the importance of not colonized suitable habitat for the conservation of epiphytes.

In contrast to the spatial motion setting, the material motion setting of continuum mechanics is concerned with the response to variations of material placements of particles with respect to the ambient material. The material motion point of view is thus extremely prominent when dealing with defect mechanics to which it has originally been introduced by Eshelby more than half a century ago. Its primary unknown, the material deformation map is governed by the material motion balance of momentum, i.e. the balance of material forces on the material manifold in the sense of Eshelby. Material (configurational) forces are concerned with the response to variations of material placements of 'physical particles' with respect to the ambient material. Opposed to that, the common spatial (mechanical) forces in the sense of Newton are considered as the response to variations of spatial placements of 'physical particles' with respect to the ambient space. Material forces as advocated by Maugin are especially suited for the assessment of general defects as inhomogeneities, interfaces, dislocations and cracks, where the material forces are directly related to the classical J-Integral in fracture mechanics, see also Gross & Seelig. Another classical example of a material - or rather configurational - force is emblematized by the celebrated Peach-Koehler force, see e.g. the discussion in Steinmann. The present work is mainly divided in four parts. In the first part we will introduce the basic notions of the mechanics and numerics of material forces for a quasi-static conservative mechanical system. In this case the internal potential energy density per unit volume characterizes a hyperelastic material behaviour. In the first numerical example we discuss the reliability of the material force method to calculate the vectorial J-integral of a crack in a Ramberg-Osgood type material under mode I loading and superimposed T-stresses. Secondly, we study the direction of the single material force acting as the driving force of a kinked crack in a geometrically nonlinear hyperelastic Neo-Hooke material. In the second part we focus on material forces in the case of geometrically nonlinear thermo-hyperelastic material behaviour. Therefore we adapt the theory and numerics to a transient coupled problem, and elaborate the format of the Eshelby stress tensor as well as the internal material volume forces induced by the gradient of the temperature field. We study numerically the material forces in a bimaterial bar under tension load and the time dependent evolution of material forces in a cracked specimen. The third part discusses the material force method in the case of geometrically nonlinear isotropic continuum damage. The basic equations are similar to those of the thermo-hyperelastic problem but we introduce an alternative numerical scheme, namely an active set search algorithm, to calculate the damage field as an additional degree of freedom. With this at hand, it is an easy task to obtain the gradient of the damage field which induces the internal material volume forces. Numeric examples in this part are a specimen with an elliptic hole with different semi-axis, a center cracked specimen and a cracked disc under pure mode I loading. In the fourth part of this work we elaborate the format of the Eshelby stress tensor and the internal material volume forces for geometrically nonlinear multiplicative elasto-plasticity. Concerning the numerical implementation we restrict ourselves to the case of geometrically linear single slip crystal plasticity and compare here two different numerical methods to calculate the gradient of the internal variable which enters the format of the internal material volume forces. The two numerical methods are firstly, a node point based approach, where the internal variable is addressed as an additional degree of freedom, and secondly, a standard approach where the internal variable is only available at the integration points level. Here a least square projection scheme is enforced to calculate the necessary gradients of this internal variable. As numerical examples we discuss a specimen with an elliptic inclusion and an elliptic hole respectively and, in addition, a crack under pure mode I loading in a material with different slip angles. Here we focus on the comparison of the two different methods to calculate the gradient of the internal variable. As a second class of numerical problems we elaborate and implement a geometrically linear von Mises plasticity with isotropic hardening. Here the necessary gradients of the internal variables are calculated by the already mentioned projection scheme. The results of a crack in a material with different hardening behaviour under various additional T-stresses are given.

Nowadays piezoelectric and ferroelectric materials are becoming more and more an interesting part of smart materials in scientific and engineering applications. Precision machining in manufacturing, micropositioning in metrology, common rail systems with piezo fuel injection control in automobile industry, and ferroelectric random access memories (FRAM) in microelectromechanical systems (MEMS) besides commercial piezo actuators and sensors can be very good examples for the application of piezoceramic and ferroelectric materials. In spite of having good characteristics, piezoelectric and ferroelectric materials have significant nonlinearities, which limit the applications in high performance usage. Domain switching (ferroelastic or ferroelectric) is the main reason for the nonlinearity of ferroelectric materials. External excessive electromechanical loads (mechanical stress and electric field) are driving forces for domain switching. In literature, various important experiments related to the non-linear properties of piezoelectric and ferroelectric materials are reported. Simulations of nonlinear properties of piezoelectric and ferroelectric materials based on physical insights of the material have been performed during the last two decades by using micromechanical and phenomenological models. The most significant experiments and models are deeply discussed in the literature survey. In this thesis the nonlinear behaviour of tetragonal perovskite type piezoceramic materials is simulated theoretically using two and three dimensional micromechanical models which are based on physical insights of the material. In the simulations a bulk piezoceramic material which has numerous grains is considered. Each grain has random orientation in properties of polarization and strain. Randomness of orientations is given by Euler angles equally distributed between \(0\) and \(2\pi\). Each element in the micromechanical model has been assumed to have the same properties of the real piezoelectric grain. In the first part of the simulations, quasi-static characteristics of piezoelectric materials are investigated by applying cyclic, rate independent, bipolar, uni-axial and external electrical loading with an amplitude of 2 kV/mm gradually starting from zero value in virgin state. Moreover, the simulations are undertaken for these materials which are subjected to quasi-static, uni-polar, uni-axial mechanical stress, namely compressive stress. The calculations are performed at each element based on linear constitutive equations, nonlinear domain switching and a probability theory for domain switching. In order to fit the simulations to the experimental data, some parameters such as spontaneous polarization, spontaneous strain, piezoelectric and dielectric constants are chosen from literature. The domain switching of each grain is determined by an electromechanical energy criterion. Depending on the actual energy related to a critical energy a certain probability is introduced for domain switching of the polarization direction. Same energy levels are assumed in the electromechanical energy relation for different types of domain switching like 90º and 180º for perovskite type tetragonal or 70.5º and 109.5º for rhombohedral microstructures. It is assumed that intergranular effects between grains can be modelled by such probability functions phenomenologically. The macroscopic response of the material to the applied electromechanical loading is calculated by using Euler transformations and averaging the individual grains. Properties of piezoelectric materials under fixed mechanical stresses are also investigated by applying constant compressive stress in addition to cyclic electrical loading in the simulations. Compressive stress is applied and kept constant before cyclic bipolar electrical loading is implemented. In the following chapters, a three-dimensional micromechanical model is extended for the simulation of the rate dependent properties of certain perovskite type tetragonal piezoelectric materials. The frequency dependent micromechanical model is now not only based on linear constitutive and nonlinear domain switching but also linear kinetics theories. The material is loaded both electrically and mechanically in separate manner with an alternating electrical voltage and mechanical stress values of various moderate frequencies, which are in the order of 0.01 Hz to 1 Hz. Electromechanical energy equation in combination with a probability function is again used to determine the onset of the domain switching inside the grains. The propagation of the domain wall during the domain switching process in grains is modelled by means of linear kinetics relations after a new domain nucleates. Electric displacement versus electric field hysteresis loops, mechanical strain versus mechanical stress and electric displacement versus mechanical stress for different frequencies and amplitudes of the alternating electric fields and compressive stresses are simulated and presented. A simple micromechanical model without using probabilistic approach is compared with the one that takes it into account. Both models give important insights into the rate dependency of piezoelectric materials, which was observed in some experiments reported in the literature. Intergranular effects are other significant factors for nonlinearities of polycrystalline ferroelectric materials. Even piezoelectric actuators and sensors show nonlinearities when they are operated with electrical loading, which is much lower than the coercive electric field level. Intergranular effects are the main cause of such small hysteresis loops. In the corresponding chapter, two basic field effects which are electrical and mechanical are taken into account for the consideration of intergranular effects micromechanically in the simulations of the two dimensional model. Therefore, a new electromechanical energy equation for the threshold of domain switching is introduced to explain nonlinearities stemming from both domain switching and intergranular effects. The material parameters like coercive electric field and critical spontaneous polarization or strain quantities are not implemented in the electromechanical energy relation. But, this relation contains new parameters which consider both mechanical and electrical field characteristics of neighbouring elements. By using this new model, mechanical strain versus electric field butterfly curves under small electrical loading conditions are also simulated. Hence, a rate dependent concept is applied in butterfly curves by means of linear kinetics model. As a result, the simulations have better matching with corresponding experiments in literature. In the next step, the model can be extended in three dimensional case and the parameters of electromechanical energy relation can be improved in order to get better simulations of nonlinear properties of polycrystalline piezoelectric materials.

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.