The submitted work deals with option pricing. Mathematical approach is immediately followed by an economic interpretation. The main problem is to model the underlying uncertainities driving the stock price. Using two well-known valuation models, binomial model and Black-Scholes model, we explain basic principles, especially risk neutral pricing. Due to the empirical biases new models have been developped, based on pure jump process. Variance gamma process and its special symmetric case are presented.

Ces dernières années ont vu apparaître sur le marché les premiers transistors de puissance de type MOSFET en carbure de silicium. Ce type de composant est particulièrement adapté à la réalisation d’équipement électrique à haut rendement et capable de fonctionner à haute température. Néanmoins, la question de la fiabilité doit être posée avant de pouvoir envisager la mise en œuvre de ces composants dans des applications aéronautiques ou spatiales. Les mécanismes de défaillance liés à l’oxyde de grille ont pendant longtemps retardé la mise sur le marché des transistors à grille isolée en carbure de silicium. Cette étude s’attache donc à estimer la durée de vie des MOSFET SiC de 1ére génération. Dans un premier temps, le mécanisme connu sous le nom de Time Dependent Dielectric Breakdown(TDDB) a été étudié au travers de résultats expérimentaux issus de la bibliographie. Notre analyse nous a permis de justifier de l’emploi d’une loi de Weibull pour modéliser la distribution des temps à défaillance issue de ces tests. Les résultats nous ont également permis de confirmer l’amélioration significative de la fiabilité de ces structures vis-à-vis de ce mécanisme. Dans un second temps, l’impact du mécanisme d’instabilité de la tension de seuil sur la fiabilité a été quantifié au travers de tests de vieillissement de type HTGB. Les données de dégradation ainsi collectées ont été modélisées à l’aide d’un processus gamma non-homogène, qui nous a permis de prendre en compte la variabilité entre les composants testés dans des conditions identiques et de proposer des facteurs d’accélération en tension et en température pour ce mécanisme. Enfin, ces travaux ont permis d’ouvrir la voie à la mise en œuvre d’outils de pronostic de la durée de vie résiduelle pour les équipements électriques. / Recent years have seen SiC MOSFET reach the industrial market. This type of device is particularly adapted to the design of power electronics equipment with high efficiency and high reliability capable to operate in high ambient temperature. Nevertheless the question of the SiC MOSFET reliability has to be addressed prior to considering the implementation of such devices in an aeronautic application. The failure mechanisms linked to the gate oxide of the SiC MOSFET have for a long time prevented the introduction of the device. In this manuscript we propose to study the reliability of the first generation of SiC MOSFET. First, the mechanism known as the Time–Dependent Dielectric Breakdown is studied through experimental results extracted from literature. Our study shows the successful application of a Weibull law to model the time-to-failure distribution extracted from the accelerated tests. The results show also a significant improvement of the SiC MOSFET structure with respect to this phenomenon. In a second step, the impact of the threshold voltage instability is quantified through accelerated tests known as High Temperature Gate Bias. The collected degradation data are modeled using a non-homogeneous Gamma process. This approach allows taking into account the variability between devices tested under the same conditions. Acceleration factors have been proposed with respect to temperature and gate voltage. Eventually the study delivers a primary estimation of the remaining useful lifetime of the SiC MOSFET in a typical aeronautic application.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.