This thesis develops a generic framework based on the Fourier transform for pricing and hedging of various options in equity, commodity, currency, and insurance markets. The pricing problem can be reduced to solving a partial integro-differential equation (PIDE). The Fourier Space Time-stepping (FST) framework developed in this thesis circumvents the problems associated with the existing finite difference methods by utilizing the Fourier transform to solve the PIDE. The FST framework-based methods are generic, highly efficient and rapidly convergent. The Fourier transform can be applied to the pricing PIDE to obtain a linear system of ordinary differential equations that can be solved explicitly. Solving the PIDE in Fourier space allows for the integral term to be handled efficiently and avoids the asymmetrical treatment of diffusion and integral terms, common in the finite difference schemes found in the literature. For path-independent options, prices can be obtained for a range of stock prices in one iteration of the algorithm. For exotic, path-dependent options, a time-stepping methodology is developed to handle barriers, free boundaries, and exercise policies. The thesis includes applications of the FST framework-based methods to a wide range of option pricing problems. Pricing of single- and multi-asset, European and path-dependent options under independent-increment exponential Levy stock price models, common in equity and insurance markets, can be done efficiently via the cornerstone FST method. Mean-reverting Levy spot price models, common in commodity markets, are handled by introducing a frequency transformation, which can be readily computed via scaling of the option value function. Generating stochastic volatility, to match the long-term equity options market data, and stochastic skew, observed in currency markets, is addressed by introducing a non-stationary extension of multi-dimensional Levy processes using regime-switching. Finally, codependent jumps in multi-asset models are introduced through copulas. The FST methods are computationally efficient, running in O(MN^d log_2 N) time with M time steps and N space points in each dimension on a d-dimensional grid. The methods achieve second-order convergence in space; for American options, a penalty method is used to attain second-order convergence in time. Furthermore, graphics processing units are utilized to further reduce the computational time of FST methods.

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Environmental and food spoilage fungi cause billions of dollars in damage in North America alone each year, in the form of rotted wood and crops, spoiled food, and human and animal illness. Each of these threats could be drastically reduced if early and more rapid detection processes are developed to replace the serological methods that are currently in practice. The current North American protocol for establishing identification of contaminating fungi both in environment and food have a time frame of approximately one week to twenty-two days. The use of a Fourier transform infrared (FTIR) spectrometer, coupled with a focal-plan-array (FPA) detector, can theoretically shorten the time (analysis within minutes after obtaining a pure culture) it takes to identify and classify a fungal cell. FPA-FTIR spectroscopy is advantageous as little to no sample preparation is required and results are obtained in less than one minute per sample. The fungal subset chosen for this study includes representatives from five phyla, including Zygomycota (Mucor heimalis), Ascomycota (Neurospora crassa, Ophiostoma minor, Chaetomium globosporum, Alternaria brassicicola), Basidiomycota (Schizophyllum commune, Chaetomium globosporum), Deutromycota (Aspergillus niger, Penicillium notatum, Aureobasidium pullulans) and the Mycetozoa (dictyostelium discoideum, physarum polycephalum). Different variables were tested and evaluated, including variability in growth parameters, wet deposition of fungi versus dry smearing of fungi, optimal absorbance range, and spectral processing parameters as well as discrepancies from one instrument to another, as well as spectral reproducibility from one instrument to another. By following the experimental protocol developed, reproducible spectra were attained, and differentiation of the fungi within the set selected for this study was achieved. The results of this work demonstrate that FPA-FTIR spectroscopy can potentially be employed for the accurate identification of environmental and food spoilage fungi.

Size-dependent material responses observed at fine length scales are receiving growing attention due to the need in the modeling of very small sized mechanical structures. The conventional continuum theories do not suffice for accurate descriptions of the exact material behaviors in the fine-scale regime due to the lack of inherent material lengths. A number of new theories/models have been propounded so far to interpret such novel phenomena. In this dissertation a few enriched-continuum theories - the adhesive contact mechanics, surface elasticity and strain gradient elasticity - are employed to study the mechanical behaviors of a semi-infinite solid induced by the boundary forces. A unified treatment of axisymmetric adhesive contact problems is developed using the harmonic functions. The generalized solution applies to the adhesive contact problems involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile, and it links existing solutions/models for axisymmetric non-adhesive and adhesive contact problems like the Hertz solution, Sneddon's solution, the JKR model, the DMT model and the M-D model. The generalized Boussinesq and Flamant problems are examined in the context of the surface elasticity of Gurtin and Murdoch (1975, 1978), which treats the surface as a negligibly thin membrane with material properties differing from those of the bulk. Analytical solution is derived based on integral transforms and use of potential functions. The newly derived solution applies to the problems of an elastic half-space (half-plane as well) subjected to prescribed surface tractions with consideration of surface effects. The newly derived results exhibit substantial deviations from the classical predictions near the loading points and converge to the classical ones at a distance far away from those points. The size-dependency of material responses is clearly demonstrated and material hardening effects are predicted. The half-space contact problems are also studied using the simplified strain gradient elasticity theory which incorporates material microstructural effects. The solution is obtained by taking advantage of the displacement functions of Mindlin (1964) and integral transforms. Significant discrepancy between the current and the classical solutions is seen to exist in the immediate vicinity of the loading area. The discontinuity and singularity exist in classical solution are removed, and the stress and displacement components change smoothly through the solid body.

The aim of this thesis is to determine the horse’s gait types using the acceleration values measured from the horse. A measurement was taken in Gävletravet, a total of five Nanotrak sensors were used, four on the different parts of the horse, and one on the hand of the horse’s driver, a car was driven parallel to the horse and the motions of the horse was recorded by a camera in order to synchronize with the data measured by the sensors, a total of four videos were recorded. The software to process the data was Matlab R2010b, and the methods to analyze them were Fast Fourier Transform (FFT), Short Time Fourier Transform (STFT), and Least Squares (LS). Different window functions were tried when applying the STFT, and the Hanning window was the best to smooth the curves, different window sizes (or data length) were also tried, the data length of 512 was found to be the most proper value. The methods for classification of horse’s gaits included amplitude, ratio, and LS. The method of amplitude worked well for the first three videos except for the last one, and performed better than the other two. The method of ratio was more reliable, but the results were not satisfactory. The method of LS gave bad results, so it was not trustworthy. More measurements and more analysis needed to be done in the future to find a proper way to automatic determine the horse’s gaits, and the use of modern technology will be very popular in other fields like animal science.

Nonlinear mesoscopic elastic (NME) materials present ananomalous nonlinear elastic behavior, which could not beexplained by classical theories. New physical mechanismsshould be individuated to explain NMEs response.Dislocations in damaged metals, fluids in rocks andadhesion (in composites) could be plausible. In this thesisI have searched for differences in the macroscopic elasticresponse of materials which could be ascribed to differentphysical processes. I have found that the nonlinearindicators follow a power law behavior as a function of theexcitation energy, with exponent ranging from 1 to 3 (thisis not completely new). This allowed to classify materialsinto well-defined classes, each characterized by a value ofthe exponent and specific microstructural properties. Tolink the measured power law exponent to plausiblephysical mechanisms, I have extended thePreisach-Mayergoyz formalism for hysteresis to multi-statemodels. Specific multi-state discrete models have beenderived from continuous microscopic physical processes,such as adhesion-clapping, adhesion-capillary forces,dislocations motion and hysteresis. In each model, themicroscopic behavior is described by a multistate equationof state, with parameters which are statisticallydistributed. Averaging over many microscopic elements theso-called mesoscopic equation of state is derived and, fromwave propagation simulations in a sample composed bymany mesoscopic elements, the experimental results couldbe reproduced. In the work of the thesis, I have shownthat model predictions of the exponent b ( the exponent bhas not been introduced before) are linked in a 'a priori'predictable way to the number of states and the propertiesof the statistical distribution adopted. We have classifiedmodels into classes defined by a different exponent b andcomparing with experimental results we have suggestedplausible mechanisms for the nonlinearity generation.

The Fast Fourier Transform (FFT) has been used in many scientific and engineering applications. The use of FFT for financial derivatives has been gaining momentum in the recent past. In this thesis, i) we have improved a recently proposed model of FFT for pricing financial derivatives to help design an efficient parallel algorithm. The improved mathematical model put forth in our research bridges a gap between quantitative approaches for the option pricing problem and practical implementation of such approaches on modern computer architectures. The thesis goes further by proving that the improved model of fast Fourier transform for option pricing produces accurate option values. ii) We have developed a parallel algorithm for the FFT using the classical Cooley-Tukey algorithm and improved this algorithm by introducing a data swapping technique that brings data closer to the respective processors and hence reduces the communication overhead to a large extent leading to better performance of the parallel algorithm. We have tested the new algorithm on a 20 node SunFire 6800 high performance computing system and compared the new algorithm with the traditional Cooley-Tukey algorithm. Option values are calculated for various strike prices with a proper selection of strike-price spacing to ensure fine-grid integration for FFT computation as well as to maximize the number of strikes lying in the desired region of the stock price. Compared to the traditional Cooley-Tukey algorithm, the current algorithm with data swapping performs better by more than 15% for large data sizes. In the rapidly changing market place, these improvements could mean a lot for an investor or financial institution because obtaining faster results offers a competitive advantages.

Three different algorithms for the generation of stationary Gaussian time series with given autocorrelation function are presented in this paper. The algorithms have already been suggested in the literature but are not well known and have never been compared before. Interrelations between the different methods, advantages and disadvantages with respect to speed and memory requirements and the range of autocorrelation functions for which the different methods are stable are discussed. The time-complexity of the algorithms and the comparisons of their implementations show that the method twice using the Fourier transform is by far the most efficient if time series of moderate or large length are generated. A tested C-code of the latter algorithm is included as this method is tricky to implement and very difficult to find in the literature. (We know only one reference, that gives a correct algorithm, but there the description is very short and no proof is included.) (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

This thesis describes the commissioning and development of a pulsed supersonic nozzle, Fourier-transform microwave spectrometer and its application to the study of several weakly bound van der Waals complexes. A pulsed supersonic expansion, Fourier-transform microwave spectrometer based on the Flygare design with a number of modifications has been constructed with an operating range of 6-18 GHz. A homodyne detection circuit mixing signals to modulus values between dc and 1 MHz is used, requiring two measurements to determine absolute transition frequencies. Transition frequencies are measured from the power spectrum by determining the first derivative zero crossing point in a least squares fitting procedure. Semiautomation of many of the spectrometer operations has been achieved allowing unattended data collection over scans of up to 300 MHz. The microwave spectrum of Ar₂-OCS and Ar₂-OC³⁴S has been observed and analysed using conventional Watson S reduction hamiltonian parameters. Effective structural parameters are derived and used in a harmonic force field analysis, based on the centrifugal distortion constants, to compare the trimer interations with a model based on the sum of dimer interactions. A series of complexes containing the nitrogen molecule undergoing tunnelling motions have been studied. Hyperfine matrix elements for the first order nuclear quadrupole interaction are derived for the coupled identical nuclei case appropriate to the rapid tunnelling motions observed. The microwave spectrum of N₂-OCS is described. Tunnelling and nuclear spin statistical effects for two symmetry states are observed arising from the interchange of nitrogen nuclei. Rotational and quadrupole constants are derived; an accidental near degeneracy of two rotational levels allows the off-diagonal quadrupole coupling constant to be determined from second order effects. A tunnelling hamiltonian fitting the quadrupole coupling constants to an angular potential has been used to calculate the tunnelling frequency and barrier to N₂ rotation. The microwave spectrum of N₂-O₃ and a preliminary spectrum of N₂-SO₃ have been observed. Rotation-inversion motions of the O₃ and SO₂ moieties must be considered in addition to the N₂ tunnelling to fit the spectrum. Tunnelling frequencies for the O₃/SO₂ and geared motions with the N₂ are derived as well as structural parameters. Modifications for production of refractory molecules and complexes by laser ablation have been made. A modified nozzle employing rods of material is used with the ablation process taking place in the nozzle throat. Modifications to obtain an expansion along the axis of the microwave cavity employ a hemispherical Fabry-Perot cavity configuration. The system has been tested on a number of diatomic molecules including PbS and CuCl.