Dans un nombre croissant d'applications, les perturbations rencontrées s'éloignent fortement des modèles classiques qui les modélisent par une gaussienne ou un mélange de gaussiennes. C'est en particulier le cas des bruits impulsifs que nous rencontrons dans plusieurs domaines, notamment celui des télécommunications. Dans ce cas, une modélisation mieux adaptée peut reposer sur les distributions alpha-stables. C'est dans ce cadre que s'inscrit le travail de cette thèse dont l'objectif est de concevoir de nouvelles méthodes robustes pour l'estimation conjointe état-bruit dans des environnements impulsifs. L'inférence est réalisée dans un cadre bayésien en utilisant les méthodes de Monte Carlo séquentielles. Dans un premier temps, cette problématique a été abordée dans le contexte des systèmes de transmission OFDM en supposant que les distorsions du canal sont modélisées par des distributions alpha-stables symétriques. Un algorithme de Monte Carlo séquentiel a été proposé pour l'estimation conjointe des symboles OFDM émis et des paramètres du bruit $\alpha$-stable. Ensuite, cette problématique a été abordée dans un cadre applicatif plus large, celui des systèmes non linéaires. Une approche bayésienne non paramétrique fondée sur la modélisation du bruit alpha-stable par des mélanges de processus de Dirichlet a été proposée. Des filtres particulaires basés sur des densités d'importance efficaces sont développés pour l'estimation conjointe du signal et des densités de probabilité des bruits / In signal processing literature, noise's sources are often assumed to be Gaussian. However, in many fields the conventional Gaussian noise assumption is inadequate and can lead to the loss of resolution and/or accuracy. This is particularly the case of noise that exhibits impulsive nature. The latter is found in several areas, especially telecommunications. $\alpha$-stable distributions are suitable for modeling this type of noise. In this context, the main focus of this thesis is to propose novel methods for the joint estimation of the state and the noise in impulsive environments. Inference is performed within a Bayesian framework using sequential Monte Carlo methods. First, this issue has been addressed within an OFDM transmission link assuming a symmetric alpha-stable model for channel distortions. For this purpose, a particle filter is proposed to include the joint estimation of the transmitted OFDM symbols and the noise parameters. Then, this problem has been tackled in the more general context of nonlinear dynamic systems. A flexible Bayesian nonparametric model based on Dirichlet Process Mixtures is introduced to model the alpha-stable noise. Moreover, sequential Monte Carlo filters based on efficient importance densities are implemented to perform the joint estimation of the state and the unknown measurement noise density

Das Konzept von aktiven Brownschen Teilchen kann benutzt werden, um das Verhalten einfacher biologischer Organismen oder künstlicher Objekte, welche die Möglichkeit besitzen sich von selbst fortzubewegen zu beschreiben. Als Bewegungsgleichungen für aktive Brownsche Teilchen kommen Langevin Gleichungen zum Einsatz. In dieser Arbeit werden aktive Teilchen mit konstanter Geschwindigkeit diskutiert. Im ersten Teil der Arbeit wirkt auf die Bewegungsrichtung des Teilchen weißes alpha-stabiles Rauschen. Es werden die mittlere quadratische Verschiebung und der effektive Diffusionskoeffizient bestimmt. Eine überdampfte Beschreibung, gültig für Zeiten groß gegenüber der Relaxationszeit wird hergleitet. Als experimentell zugängliche Meßgröße, welche als Unterscheidungsmerkmal für die unterschiedlichen Rauscharten herangezogen werden kann, wird die Kurtose berechnet. Neben weißem Rauschen wird noch der Fall eines Ornstein-Uhlenbeck Prozesses angetrieben von Cauchy verteiltem Rauschen diskutiert. Während eine normale Diffusion mit zu weißem Rauschen identischem Diffusionskoeffizienten bestimmt wird, kann die beobachtete Verteilung der Verschiebungen Nicht-Gaußförmig sein. Die Zeit für den Übergang zur Gaußverteilung kann deutlich größer als die Zeitskale Relaxationszeit und die Zeitskale des Ornstein-Uhlenbeck Prozesses sein. Eine Grenze der benötigten Zeit wird durch eine Näherung der Kurtosis ermittelt. Weiterhin werden die Grundlagen eines stochastischen Modells für lokale Suche gelegt. Lokale Suche ist die Suche in der näheren Umgebung eines bestimmten Punktes, welcher Haus genannt wird. Abermals diskutieren wir ein aktives Teilchen mit unveränderlichem Absolutbetrag der Geschwindigkeit und weißen alpha-stabilem Rauschen in der Bewegungsrichtungsdynamik. Die deterministische Bewegung des Teilchens wird analysiert bevor die Situation mit Rauschen betrachtet wird. Die stationäre Aufenthaltswahrscheinlichkeitsdichtefunktion wird bestimmt. Es wird eine optimale Rauschstärke für die lokale Suche, das heißt für das Auffinden eines neuen Ortes in kleinstmöglicher Zeit festgestellt. Die kleinstmögliche Zeit wird kaum von der Rauschart abhängen. Wir werden jedoch feststellen, dass die Rauschart deutlichen Einfluß auf die Rückkehrwahrscheinlichkeit zum Haus hat, wenn die Richtung des zu Hauses fehlerbehaftet ist. Weiterhin wird das Model durch eine an das Haus abstandsabhängige Kopplung erweitert werden. Zum Abschluß betrachten wir eine Gruppe von Suchern. / Active Brownian particles described by Langevin equations are used to model the behavior of simple biological organisms or artificial objects that are able to perform self propulsion. In this thesis we discuss active particles with constant speed. In the first part, we consider angular driving by white Levy-stable noise and we discuss the mean squared displacement and diffusion coefficients. We derive an overdamped description for those particles that is valid at time scales larger the relaxation time. In order to provide an experimentally accessible property that distinguishes between the considered noise types, we derive an analytical expression for the kurtosis. Afterwards, we consider an Ornstein-Uhlenbeck process driven by Cauchy noise in the angular dynamics of the particle. While, we find normal diffusion with the diffusion coefficient identical to the white noise case we observe a Non-Gaussian displacement at time scales that can be considerable larger than the relaxation time and the time scale provided by the Ornstein-Uhlenbeck process. In order to provide a limit for the time needed for the transition to a Gaussian displacement, we approximate the kurtosis. Afterwards, we lay the foundation for a stochastic model for local search. Local search is concerned with the neighborhood of a given spot called home. We consider an active particle with constant speed and alpha-stable noise in the dynamics of the direction of motion. The deterministic motion will be discussed before considering the noise to be present. An analytical result for the steady state spatial density will be given. We will find an optimal noise strength for the local search and only a weak dependence on the considered noise types. Several extensions to the introduced model will then be considered. One extension includes a distance dependent coupling towards the home and thus the model becomes more general. Another extension concerned with an erroneous understanding by the particle of the direction of the home leads to the result that the return probability to the home depends on the noise type. Finally we consider a group of searchers.

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 ).

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 ).