<p>Print quality (PQ) is one of the most significant issues with electrophotographic printers. There are many reasons for PQ issues, such as limitations of the electrophotographic process, faulty printer components, or other failures of the print mechanism. These reasons can produce different PQ issues, like streaks, bands, gray spots, text fading, and color fading defects. It is important to analyze the nature and causes of different print defects to more efficiently repair printers and improve the electrophotographic process. </p> <p><br></p> <p>We design a general framework for print quality detection and analysis of customer content. This print quality analysis framework inputs the original digital image saved on the computer and then the scanned image. This framework includes two main modules: image pre-processing, print defects feature vector extraction, and classification. The first module, image pre-processing, includes image registration, color calibration, and region of interest (ROI) extraction. The ROI extraction part is designed to extract four different kinds of ROI from the digital master image. Because different ROIs include different print defects, for example, the symbol ROI includes the text fading defect, and the raster ROI includes the color fading defect. The second module includes different ROI print defects detection and analysis algorithms. We classify different ROI print defects using their feature vector based on their severity. This module proposed four important defects detection methods: uniform color area streak detection, symbol ROI color text fading detection, raster ROI color fading detection using a novel unsupervised clustering method, and raster ROI streak detection. We will introduce the details of these algorithms in this thesis. </p> <p><br></p> <p>We will also show two other interesting print quality projects: print margin skew detection and print velocity simulation and estimation. Print margin skew detection proposes an algorithm that uses the Hough Lines Detection algorithm to detect printing margin and skew errors based on factual scanned image verification. In the print velocity simulation and estimation project, we propose a print velocity simulation tool, design a specific print velocity test page, and design a print velocity estimation algorithm using the dynamic time warping algorithm. </p>

Concepts in optimal search theory have been used in human-based aerial search since World War II. This thesis addresses the technical and theoretical issues necessary to apply this crucial theory to search path planning for Small Unmanned Aerial Vehicles (SUAVs). A typical search often requires that more than one target be located. Accordingly, a method is presented to locate multiple targets in three dimensions, as well as to differentiate between them. However, significant error can be present when locating targets from an airborne platform, and the idea of target quality is also introduced as a way to describe the reliability of target estimates. Flight test results are presented to validate the target differentiation algorithm. In this test, five out of six targets as close as 6.1 m apart are located and differentiated with less than four meters of error. This flight test also provides color information that is useful in generating artificial target images and understanding the target detection probability. Image skew is then incorporated into the detection probability model, and a function is derived that predicts target detection as a function of distance. In order to measure the effectiveness of search algorithms with this model, the concept of a probability map is introduced. This map can be updated as the search progresses, and is stored on a probability grid containing nodes that keep track of the probable target locations and the probability of detection. Using this tool, a search width is developed for a given airborne agent. The search width is then used to derive optimal search performance based on a given probability map and SUAV. Finally, the concepts of efficiency and completeness are given specific definitions in the context of discrete search. These metrics are used to develop a search plan that focuses on efficiency, and one that focuses on completeness. Example simulations are used to illustrate the conditions under which each plan might be desirable, and a composite search strategy is presented that combines both plans.

In the last decade, a considerable growth has been added to the volume of the credit risk derivatives market. This growth has been followed by the current financial market turbulence. These two periods have outlined how significant and important are the credit derivatives market and its products. Modelling-wise, this growth has parallelised by more complicated and assembled credit derivatives products such as mth to default Credit Default Swaps (CDS), m out of n (CDS) and collateralised debt obligation (CDO). In this thesis, the Lévy process has been proposed to generalise and overcome the Credit Risk derivatives standard pricing model's limitations, i.e. Gaussian Factor Copula Model. One of the most important drawbacks is that it has a lack of tail dependence or, in other words, it needs more skewed correlation. However, by the Lévy Factor Copula Model, the microscopic approach of exploring this factor copula models has been developed and standardised to incorporate an endless number of distribution alternatives those admits the Lévy process. Since the Lévy process could include a variety of processes structural assumptions from pure jumps to continuous stochastic, then those distributions who admit this process could represent asymmetry and fat tails as they could characterise symmetry and normal tails. As a consequence they could capture both high and low events¿ probabilities. Subsequently, other techniques those could enhance the skewness of its correlation and be incorporated within the Lévy Factor Copula Model has been proposed, i.e. the 'Stochastic Correlated Lévy Factor Copula Model' and 'Lévy Random Factor Loading Copula Model'. Then the Lévy process has been applied through a number of proposed Pricing Basket CDS&CDO by Lévy Factor Copula and its skewed versions and evaluated by V-FFT limiting and mixture cases of the Lévy Skew Alpha-Stable distribution and Generalized Hyperbolic distribution. Numerically, the characteristic functions of the mth to default CDS's and (n/m) th to default CDS's number of defaults, the CDO's cumulative loss, and loss given default are evaluated by semi-explicit techniques, i.e. via the DFT's Fast form (FFT) and the proposed Very Fast form (VFFT). This technique through its fast and very fast forms reduce the computational complexity from O(N2) to, respectively, O(N log2 N ) and O(N ).

No description available.

No description available.

A Growth Curve Model (GCM) is a multivariate linear model used for analyzing longitudinal data with short to moderate time series. It is a special case of Generalized Multivariate Analysis of Variance (GMANOVA) models. Analysis using the GCM involves comparison of mean growths among different groups. The classical GCM, however, possesses some limitations including distributional assumptions, assumption of identical degree of polynomials for all groups and it requires larger sample size than the number of time points. In this thesis, we relax some of the assumptions of the traditional GCM and develop appropriate inferential tools for its analysis, with the aim of reducing bias, improving precision and to gain increased power as well as overcome limitations of high-dimensionality. Existing methods for estimating the parameters of the GCM assume that the underlying distribution for the error terms is multivariate normal. In practical problems, however, we often come across skewed data and hence estimation techniques developed under the normality assumption may not be optimal. Simulation studies conducted in this thesis, in fact, show that existing methods are sensitive to the presence of skewness in the data, where estimators are associated with increased bias and mean square error (MSE), when the normality assumption is violated. Methods appropriate for skewed distributions are, therefore, required. In this thesis, we relax the distributional assumption of the GCM and provide estimators for the mean and covariance matrices of the GCM under multivariate skew normal (MSN) distribution. An estimator for the additional skewness parameter of the MSN distribution is also provided. The estimators are derived using the expectation maximization (EM) algorithm and extensive simulations are performed to examine the performance of the estimators. Comparisons with existing estimators show that our estimators perform better than existing estimators, when the underlying distribution is multivariate skew normal. Illustration using real data set is also provided, wherein Triglyceride levels from the Framingham Heart Study is modelled over time. The GCM assumes equal degree of polynomial for each group. Therefore, when groups means follow different shapes of polynomials, the GCM fails to accommodate this difference in one model. We consider an extension of the GCM, wherein mean responses from different groups can have different shapes, represented by polynomials of different degree. Such a model is referred to as Extended Growth Curve Model (EGCM). We extend our work on GCM to EGCM, and develop estimators for the mean and covariance matrices under MSN errors. We adopted the Restricted Expectation Maximization (REM) algorithm, which is based on the multivariate Newton-Raphson (NR) method and Lagrangian optimization. However, the multivariate NR method and hence, the existing REM algorithm are applicable to vector parameters and the parameters of interest in this study are matrices. We, therefore, extended the NR approach to matrix parameters, which consequently allowed us to extend the REM algorithm to matrix parameters. The performance of the proposed estimators were examined using extensive simulations and a motivating real data example was provided to illustrate the application of the proposed estimators. Finally, this thesis deals with high-dimensional application of GCM. Existing methods for a GCM are developed under the assumption of ‘small p large n’ (n >> p) and are not appropriate for analyzing high-dimensional longitudinal data, due to singularity of the sample covariance matrix. In a previous work, we used Moore-Penrose generalized inverse to overcome this challenge. However, the method has some limitations around near singularity, when p~n. In this thesis, a Bayesian framework was used to derive a test for testing the linear hypothesis on the mean parameter of the GCM, which is applicable in high-dimensional situations. Extensive simulations are performed to investigate the performance of the test statistic and establish optimality characteristics. Results show that this test performs well, under different conditions, including the near singularity zone. Sensitivity of the test to mis-specification of the parameters of the prior distribution are also examined empirically. A numerical example is provided to illustrate the usefulness of the proposed method in practical situations. / Thesis / Doctor of Philosophy (PhD)

<p dir="ltr">Rolling element bearings (REBs) are common components in rotating equipment. They are used to carry loads and allow for rotation and misalignments with minimal friction. There exists a wide variety of ball and roller bearings that are suited for a wide variety of applications. All varieties of REBs operate with the same fundamental principles: force transferred from the shaft is applied to the inner race of a bearing, distributed among the rolling elements, and passed on through the outer race to the bearing housing. Load distribution among the rolling elements and the dynamic performance of the bearing is dependent on the bearing’s specifications and operating conditions. Bearing-housing and inner race-shaft fit classifications also control the bearing radial internal clearance (RIC), which eventually affects the bearing performance and load transferred to the housing.</p><p dir="ltr">This thesis experimentally and analytically investigates the load distribution and dynamic performance of rolling elements and investigates roller slip, tilt, and skew in a spherical roller bearing (SRB) under various combinations of loads and speeds. In order to have better insight into the effect of flexible housing and shaft on load distribution and dynamics of REBs, it was experimentally investigated the variation of inner race-shaft and outer race-housing interfaces on load and pressure maps at the bearing-housing interface for four different varieties of rolling element bearing: deep groove ball bearings, angular contact ball bearings, cylindrical roller bearings, and spherical roller bearings. Moreover, an integrated rotor-bearing housing system model developed to examine the behaviors of the rotor, bearing, and housing operating under various conditions.</p><p dir="ltr">In order to gain a deeper understanding of the dynamic behavior of REBs, a full six degree of freedom SRB dynamic model was developed in MSC ADAMS software. C++ based ADAMS/Solver subroutines, called dynamic bearing model (DBM), were developed and incorporated in ADAMS to compute reaction forces and moments in a rolling element bearing. DBM is based on the discrete element method (DEM), which assumes each of the bearing elements (i.e., rolling elements, cage, inner race, and outer race) to be a rigid body with six degrees-of freedom (DOF) in a three-dimensional space. A novel test rig (spherical roller bearing test rig, SRBTR) was also designed and developed to investigate load distribution and roller slip, tilt, and skew in an SRB. The test rig utilized a double-row SRB and was designed to allow for direct visual access to each row using a high-speed camera. The dynamic behavior of the rollers was corroborated with the developed analytical model. The experimental and analytical results indicate that the roller tilt angle increases with axial load, remains constant with speed, and decreases with increasing radial load when the roller is located in the load zone. Furthermore, roller skew in the load zone increases with axial load and shaft speed; however, it decreases with the radial load. The results indicate that when the radial-to-axial load ratio is greater than 4, roller tilt and skew are minimized. Due to roller intermittent slip and roller cage pocket collision in the unload zone, tilt and skew become unpredictable. The magnitude of the tilt and skew in the unload zone is directly related to the roller-race and roller-cage pocket clearances, respectively. Another test rig (pressure mapping test rig, PMTR) was designed to solely investigate how bearing-housing and inner race-shaft interfaces affect the load distribution in REBs. Thin film pressure sensors were utilized and placed around the perimeter of the test bearings inside of a housing to experimentally evaluate the pressure distribution between REBs and a housing under different loads and bearing-shaft and bearing-housing interfaces. Pressure map results were used to evaluate the effect of radial internal clearance on the load distribution of different bearing types. Pressure map results confirmed that the amplitude of load variation reduces with the bearing internal clearance. The thin film sensor system was also used to investigate the circumferential load distribution on the housing.</p><p dir="ltr">Previous ADAMS bearing models have assumed the bearing outer race to be fixed to the ground and the bearing inner race to be attached to a rigid shaft. In order to develop a more realistic and versatile bearing simulation tool, ADAMS bearing models were combined with flexible housings and rotor. To achieve an integrated rotor-bearing housing system model, the ADAMS bearing model was coupled through a set of interface points using component-mode-synthesis (CMS) for the rotor and housing model. The bearing outer races were discretized into multiple nodes to compute the force and deformation at the bearing housing conformal contact as well as to minimize the computational requirements associated with the conformal contact problems. The integrated model was then utilized to investigate the effects of rotor flexibility in the bearing rotor system and the effect of bearing clearance and housing clearance on bearing dynamics. It was demonstrated that the flexibility of the rotor has a significant effect on bearing element motion and dynamics. The results also indicated that depending on the bearing type, the shaft deflection can induce a moment within the bearing that is not readily identifiable from elementary theory. The results showed that the flexible housing undergoes deformations that create ovality in the bearing housing, thus affecting bearing dynamics. The model was also used to investigate bearing performance in a miniature wind turbine main shaft, utilizing a combination of SRB and cylindrical roller bearing (CRB) ADAMS models. Results suggest that the axial-to-radial load ratio should be less than the tangent of the SRB contact angle to avoid premature failure due to rollers sliding in the SRB as well as detrimental parallel misalignment in the CRB.</p>

Der Einfluß der mütterlichen Verwandtschaft auf das soziale Verhalten ist eingehend für viele Primatenarten untersucht worden, es ist jedoch schwierig die Bedeutung der Verwandtenselektion auf die Evolution von sozialem Verhalten zu bewerten, wenn Studien auf die mütterliche Verwandtschaft begrenzt sind und die väterliche Verwandtschaft dabei völlig ignorieren. Das Ziel der vorliegenden Dissertation war die Untersuchung des Ausmaßes der väterlichen Verwandtschaft und ihr Einfluß auf die sozialen Beziehungen zwischen adulten Weibchen in einer freilebenden Gruppe von Rhesusaffen (Macaca mulatta) auf der Insel Cayo Santiago. Die wichtigsten Resultate können folgendermaßen zusammengefaßt werden: Erstens, zur Beurteilung des Ausmaßes der väterlichen Verwandtschaft wurde die Vaterschaft der zwischen 1993 und 1998 geborenen Kindern der Studiengruppe bestimmt. Die Resultate zeigten, dass die Reproduktion der Männchen innerhalb der Untersuchungszeit nicht gleich verteilt war, einige wenige Männchen zeugten viele Nachkommen, aber die Mehrzahl der potentiellen Väter reproduzierten sich gar nicht oder nur wenig. Dies führte zu einer Verwandtschaftsstruktur in der fast alle im Untersuchungszeitraum geborenen Kinder mindestens ein väterliches Halbgeschwister mit geringer Altersdifferenz hatten, d.h. entweder im selben Alter (74%) oder maximal zwei Jahre jünger oder älter als es selbst (15%). Wenn auch bei anderen Primatenarten die Reproduktion auf einige wenige Männchen beschränkt ist, dann sollte die Bedeutung der väterlichen Verwandtschaft für Primaten neu bewertet werden. Eine der wichtigsten Konsequenzen der ungleichen Reproduktion der Männchen wäre nämlich, dass viele Individuen mehr väterliche, als mütterliche Halbgeschwister in ihrem Leben haben. Zweitens, um den Einfluß der väterlichen Verwandtschaft auf die sozialen Beziehungen unter adulten Weibchen zu bewerten, wurden affiliative und aggressive Interaktionen von 34 Fokusweibchen und ihren sozialen Partnern beobachtet, die entweder ihre mütterlichen Halbschwestern, ihre väterlichen Halbschwestern oder nichtverwandte Weibchen waren. Die vorliegende Untersuchung bestätigte, dass mütterliche Halbschwestern die engsten affiliativen Beziehungen haben. Das wahrscheinlich wichtigste Resultat dieser Studie ist der Befund, dass adulte Weibchen deutlich häufiger affiliativ mit ihren väterlichen Halbschwestern als mit Nichtverwandten interagierten. Die Erkennung der väterlichen Halbschwestern war deutlicher unter Weibchen gleichen Alters als unter Weibchen unterschiedlichen Alters, wobei die affiliativen Interaktionen mit zunehmender Altersdifferenz (gemessen in Jahren) abnahmen. Dies deutet darauf hin, dass Altersnähe einen zusätzlichen regulierenden Einfluß auf affiliatives Verhalten hatte. Beweise dafür, dass sich väterliche Verwandte erkennen können, sind nur in Bezug auf affiliative Interaktionen, nicht aber auf dyadische Aggression gefunden worden, was eine kontextabhängige Verwandtenerkennung annehmen läßt. Drittens, bei der Einbeziehung weiterer Verwandtschaftskategorien zeigten die Daten, dass mütterliche Verwandte auch bei konstant gehaltenem Verwandtschaftsgrad deutlich gegenüber väterlichen Verwandten bevorzugt wurden. Dies weist darauf hin, dass die mütterliche Verwandtschaft in der untersuchten Gruppe einen größeren Einfluß auf die sozialen Beziehungen adulter Rhesusaffenweibchen hatte als die väterliche Verwandtschaft. Affiliative sowie aggressive Interaktionen nahmen mit sinkendem Verwandtschaftsgrad ab, aber entfernte Verwandte unterschieden einander immer noch von Nichtverwandten, was der Existenz einer Verwandschaftsschwelle widerspricht. Viertens, schließlich wurde die väterliche Verwandtschaft in Bezug auf Koalitionsbildungen untersucht. Eine Koalition wird gebildet, wenn ein Individuum in ein laufenden Konflikt zwischen zwei Kontrahenten eingreift, um eine Partei gegen die andere zu unterstützen. Weibliche Rhesusaffen unterstützten am häufigsten ihre mütterliche Halbschwestern. Außerdem unterstützten sie Nichtverwandte gleichen Alters häufiger als Nichtverwandte unterschiedlichen Alters. Weibchen unterstützten ihre väterlichen Halbschwestern nicht häufiger als Nichtverwandte, aber die Daten könnten auf einen indirekten Beweis der väterlichen Verwandtenerkennung hinweisen, da Weibchen dazu tendierten gegen ihre väterlichen Halbschwestern seltener zu intervenieren als gegen Nichtverwandte. Dieser Befund könnte durch die Tatsache bedingt sein, dass väterliche Halbschwestern sehr unterschiedliche Dominanzränge einnehmen können, währenddessen mütterliche Halbschwestern immer benachbarte Dominanzränge haben, was zur Folge hat, dass ein rangtiefes Weibchen ihrer väterlichen Halbschwester nicht wirklich helfen kann, da sie ein viel höheres Vergeltungsrisiko auf sich nimmt, wenn sie in einen Konflikt zwischen zwei ranghöhere Kontrahenten interveniert. Als einen Kompromiß könnten Weibchen statt dessen vermeiden gegen ihre väterlichen Halbschwestern zu intervenieren, was darauf hindeuten könnte, dass individuelle Zwänge bezüglich der eigenen Konkurrenzfähigkeit eine bedeutende Rolle bei der Koalitionsbildung spielen. Mütterliche und väterliche Halbschwestern zeigten zudem einen stärkeren Trend zu "reciprocity" und "interchange" als Nichtverwandte und gaben einander einen höheren Anteil an kostenintensiven Unterstützungen. Schließlich unterstreichen die Ergebnisse dieser Studie deutlich, dass Vertrautheit zwischen Individuen durch frühkindliche Bindungen mindestens auf zwei Wegen erfolgen kann: (i) Mütter vermitteln Vertrautheit zwischen ihren Kindern (die mütterliche Halbgeschwister sind) aufgrund der durch die Laktation bedingten engen Mutter-Kind Bindung und (ii) Altersnähe vermittelt mehr Vertrautheit zwischen Altersgenossen (die entweder väterliche Halbgeschwister oder Nichtverwandte sind), die wichtige Lebensabschnitte wie z.B. Kindheit, Menstruationsbeginn, Schwangerschaft oder Mutterschaft zu ähnlichen Zeiten durchlaufen, während Weibchen unterschiedlichen Alters dies nicht tun. In jedem Fall brauchen väterliche Halbgeschwister neben der Vertrautheit zu ihren Altersgenossen noch einen zusätzlichen Mechanismus wie z.B. das "phenotype matching" um innerhalb ihrer Altersgenossen wirklich zwischen väterlichen Halbgeschwistern und Nichtverwandten zu unterscheiden. / The impact of maternal kinship on social behaviour has been studied in detail for many primate species, but it is difficult to assess the importance of kin selection in shaping the evolution of social behaviour when studies are limited to maternal kin, completely ignoring paternal kinship. This thesis aimed to investigate the extent of paternal kinship and its impact on the social relationships among adult females in one group of free-ranging rhesus macaques (Macaca mulatta) living on the island of Cayo Santiago. The main findings can be summarised as follows: Firstly, in order to access the extent of paternal kinship, paternity has been analysed for all infants of the study group born between 1993 and 1998. Results revealed that male reproduction was highly skewed over the study period as few males have sired a high number of offspring, but the majority of potential sires have sired no or few offspring. This created a kinship structure in which nearly all animals born during the study period had at least one paternal half-sibling in close age proximity, i.e., either of the same age (74%) or within a two-year age difference of themselves (15%). Assuming that male reproductive success is also skewed in other primate species, then the importance of paternal kinship in primate societies should be re-emphasised. One of the most important consequences of male reproductive skew is that many individuals will have more paternal than maternal half-siblings during their life time. Secondly, in order to access the impact of paternal kinship on the social relationships among adult females, focal data on affiliation and aggression have been collected on 34 adult females with respect to their social partners who were either their maternal half-sisters, paternal half-sisters or unrelated females. The present study confirmed that the closest affiliative relationships characterise maternal half-sisters. Probably the most important result of this study was the finding that adult females were significantly more affiliative with their paternal half-sisters than with their non-kin. The recognition of paternal sisters was more pronounced among females of the same age than among females of different age, with a decrease in affiliation as the exact age difference (measured in years) increased among paternal half-sisters. This indicates that age proximity had an additional regulatory effect upon affiliative behaviour. However, evidence for paternal kin discrimination was only found with respect to affiliation, but not with respect to dyadic aggression suggesting context-dependent kin discrimination. Thirdly, when more kin categories were included in the analysis, adult females showed a strong bias towards maternal kin in comparison to paternal kin. This bias towards maternal kin when the degree of relatedness was held constantly suggests, that maternal kinship had a larger impact on the social relationships among adult female rhesus macaques than paternal kinship at least in the study group. Both affiliation and aggression declined with decreasing degrees of relatedness, but distant kin still differentiated each other from non-kin contradicting the existence of a relatedness threshold. Fourthly, paternal kinship was finally investigated with respect to coalition formation. A coalition is formed when an individual intervenes in an ongoing conflict between two opponents in order to support one party against the other. Female rhesus macaques intervened most often on behalf of their maternal half-sisters. In addition, unrelated female peers supported each other more often than unrelated female non-peers. Females did not support their paternal half-sisters more often than non-kin, but data may indicate indirect evidence for paternal kin discrimination as females tended to target their paternal half-sisters less often than non-kin. This finding might be due to the fact, that paternal half-sisters can be very different in rank, while maternal half-sisters are of adjacent rank, implying that a low-ranking female cannot provide actual help to her paternal half-sisters, as she may risk a higher probability of retaliation when intervening in a conflict between two higher-ranking opponents. As a compromise, females may instead avoid to target their paternal half-sisters, suggesting that constraints to an individuals own competitive ability play an important role in coalition formation. Maternal and paternal half-sisters showed a stronger trend in reciprocity and interchange than non-kin and also provided a higher proportion of costly interventions towards each other. Finally, the results of the present study strongly suggest that familiarity among individuals can arise through association in early development by at least two alternatives: (i) mothers mediating familiarity among their offspring (which are maternal half-siblings) caused by the close mother-offspring relationship during lactation and (ii) age proximity is mediating familiarity among age mates (including both paternal related and unrelated peers) as peers go through important life history stages such as infancy, menarche, pregnancy or motherhood at similar times while females of different age do not. In any case, paternal half-siblings additionally need a mechanism such as phenotype matching to discriminate paternal half-siblings from non-kin even within their peer group.

La tesi comprende tre saggi sul ruolo della volatilità implicita per le banche. La tesi è organizzata in tre capitoli. Capitolo I - studia il ruolo di skew e spread della volatilità implicita nel determinare i rendimenti delle azioni bancarie. Capitolo II - analizza gli effetti degli skew della volatilità implicita e della realized volatility sulla leva finanziaria delle banche. Capitolo III - si focalizza sul rapporto tra il coefficiente di liquidità delle banche e le misure per il rischio estratte dalla volatilità (skew, spread, realized volatility). / The thesis comprehends three essays on option implied volatility risk measures for banks. The thesis is organized in three chapters. Chapter I - studies the informational content for banks' stock returns in option's implied volatilities skews and spread. Chapter II - analyzes the effect of volatility risk measures (volatility skew and realized volatility) on banks' leverage. Chapter III - studies the relationship between banks' liquidity ratio and volatility risk measures.

In the last decade, a considerable growth has been added to the volume of the credit risk derivatives market. This growth has been followed by the current financial market turbulence. These two periods have outlined how significant and important are the credit derivatives market and its products. Modelling-wise, this growth has parallelised by more complicated and assembled credit derivatives products such as mth to default Credit Default Swaps (CDS), m out of n (CDS) and collateralised debt obligation (CDO). In this thesis, the Lévy process has been proposed to generalise and overcome the Credit Risk derivatives standard pricing model's limitations, i.e. Gaussian Factor Copula Model. One of the most important drawbacks is that it has a lack of tail dependence or, in other words, it needs more skewed correlation. However, by the Lévy Factor Copula Model, the microscopic approach of exploring this factor copula models has been developed and standardised to incorporate an endless number of distribution alternatives those admits the Lévy process. Since the Lévy process could include a variety of processes structural assumptions from pure jumps to continuous stochastic, then those distributions who admit this process could represent asymmetry and fat tails as they could characterise symmetry and normal tails. As a consequence they could capture both high and low events' probabilities. Subsequently, other techniques those could enhance the skewness of its correlation and be incorporated within the Lévy Factor Copula Model has been proposed, i.e. the 'Stochastic Correlated Lévy Factor Copula Model' and 'Lévy Random Factor Loading Copula Model'. Then the Lévy process has been applied through a number of proposed Pricing Basket CDS&CDO by Lévy Factor Copula and its skewed versions and evaluated by V-FFT limiting and mixture cases of the Lévy Skew Alpha-Stable distribution and Generalized Hyperbolic distribution. Numerically, the characteristic functions of the mth to default CDS's and (n/m) th to default CDS's number of defaults, the CDO's cumulative loss, and loss given default are evaluated by semi-explicit techniques, i.e. via the DFT's Fast form (FFT) and the proposed Very Fast form (VFFT). This technique through its fast and very fast forms reduce the computational complexity from O(N2) to, respectively, O(N log2 N ) and O(N ).