Ce travail s'intéresse à l'utilisation de la méthode de quasi-réversibilité pour la résolution de problèmes inverses, un exemple typique étant le problème inverse de l'obstacle. Nous proposons pour ce dernier une nouvelle approche couplant la méthode de quasi-réversibilité et une méthode de lignes de niveau. Plus précisément, à partir d'un ouvert candidat C, nous résolvons un problème de Cauchy à l'extérieur de C, puis nous mettons à jour cet ouvert par la méthode de lignes de niveau. La solution approchée du problème de Cauchy est obtenue en utilisant la méthode de quasi-réversibilité, introduite par J.L. Lions et R. Lattès dans les années soixante. Nous proposons différentes formulations de cette méthode, ainsi que sa discrétisation par éléments finis non conformes adaptés à l'espace de Sobolev H2, et nous prouvons la convergence des éléments finis. En présence d'une donnée bruitée, nous introduisons une nouvelle méthode basée sur la dualité en optimisation et le principe de Morozov. Nous montrons que cette méthode fournit des données régularisées et un choix de paramètre de régularisation pertinent pour la quasi-réversibilité. En ce qui concerne la mise à jour de l'ouvert C, nous proposons deux méthodes de lignes de niveau très différentes : la première est basée sur une équation eikonale, la seconde sur une équation de Poisson. Nous prouvons que ces deux approches assurent la convergence vers l'obstacle. Finalement, nous présentons des résultats numériques pour cette approche couplant quasi-réversibilité/lignes de niveau dans différentes situations : problème inverse de l'obstacle avec condition de Dirichlet, détection de défauts dans une structure élasto-plastique...

L'étude présentée dans ce travail de thèse aborde deux aspects du problème de Cauchy caractéristique en relativité générale. D'une part, une formule intégrale pour le problème de Cauchy caractéristique pour l'équation de Dirac est établie, généralisant les travaux de Penrose en espace-temps courbe. Ayant adapté le cadre fonctionnel pour obtenir une théorie des distributions adaptée à la structure algébriques des spineurs, le formalisme Geroch-Held-Penrose est utilisé pour décrire de la manière la plus précise possible la formule intégrale. La formule de Penrose en spin arbitraire sur l'espace-temps de Minkowski est retrouvée. D'autre part, une théorie de scattering conforme pour une équation des ondes non linéaire conformément invariante sur un espace asymptotiquement simple est construite. En effectuant un rééchelonnement conforme, l'espace-temps est complété en lui ajoutant une frontière constituée de deux hypersurfaces caractéristiques représentant respectivement les extrémités passées et futures des géodésiques de type lumière. Le comportement asymptotique des champs s'obtient alors en considérant les traces des solutions de l'équation conforme sur ces bords. L'inversibilité des opérateurs de trace s'obtient alors en résolvant un problème de Cauchy caractéristique sur ce bord et l'opérateur de scattering conforme par composition de ces opérateurs de trace.

This thesis consists of two parts: a synthesis of the theory of categories enriched in a quantaloid; and a weakening of this theory for it to include semicategories describing ordered sheaves on a quantaloid. A synthesis of, and supplements to, results in the literature concerning the theory of categories enriched in a quantaloid Q (as particular case of categories enriched in a bicategory) is contained in the first chapters. This theory is built with Q-categories, functors and distributors, and contains such notions as, for example, adjoint functors, weighted colimits, presheaves, Kan extensions, Cauchy completions and Morita equivalence, and so on. The literature does not provide an overview of these matters, so it was necessary to provide one here. Then the necessary theory is developed to arrive at an elementary description of ``ordered sheaves on a quantaloid Q', henceforth referred to as Q-orders. As there is no ``topos of sheaves on a quantaloid', Q-orders cannot be defined as ordered objects in such a topos. Instead a description of Q-orders as categorical structures enriched in the quantaloid Q is proposed. The well-known ordered sheaves on a locale L (i.e.~ordered objects in the topos of sheaves on L) should of course be a particular example of the general theory, taking Q to be the (one-object suspension of) L. Then it turns out that the theory of Q-categories has to be weakened to include ``categories without units', i.e. Q-semicategories. But for Q-semicategories to admit a convenient distributor calculus, a ``regularity' condition has to be imposed. And for those regular Q-semicategories to admit a reasonable theory of Cauchy completions and Morita equivalence, the even stronger condition of ``total regularity' has to be imposed. The former notion has been studied before for semicategories enriched in a symmetric monoidal closed category; the latter notion is new, and is introduced via the intuitively clear idea of ``stability of objects'. The point is then that precisely the Cauchy complete totally regular Q-semicategories are the Q-orders; for a locale L they are indeed the ordered objects in the topos of sheaves on L. A (bi)equivalent description of those Q-orders can be given in terms of categories enriched in the split-idempotent completion of the quantaloid Q: a totally regular semicategory enriched in Q corresponds in a precise sense to a category enriched in the split-idempotent completion of Q. Applying this once more to a locale L instead of a quantaloid Q, these results thus deepen the work of the Louvain-la-Neuve school, and reconcile it with that of the Sydney school, on the description of (ordered) sheaves on a locale as enriched categorical structures. The extended introduction gives a compact yet intuitive presentation of the developments contained in the thesis.

This dissertation collects two papers regarding the econometric and economic theory and testing of the predictability of asset returns. It is widely accepted that stock returns are not only predictable but highly so. This belief is due to an abundance of existing empirical literature fi nding often overwhelming evidence in favor of predictability. The common regressors used to test predictability (e.g., the dividend-price ratio for stock returns) are very persistent and their innovations are highly correlated with returns. Persistence when combined with a correlation between innovations in the regressor and asset returns can cause substantial over-rejection of a true null hypothesis. This result is both well documented and well known. On the other hand, stochastic volatility is both broadly accepted as a part of return time series and largely ignored by the existing econometric literature on the predictability of returns. The severe e ffect that stochastic volatility can have on standard tests are demonstrated here. These deleterious e ffects render standard tests invalid. However, this problem can be easily corrected using a simple change of chronometer. When a return time series is read in the usual way, at regular intervals of time (e.g., daily observations), then the distribution of returns is highly non-normal and displays marked time heterogeneity. If the return time series is, instead, read according to a clock based on regular intervals of volatility, then returns will be independent and identically normally distributed. This powerful result is utilized in a unique way in each chapter of this dissertation. This time-deformation technique is combined with the Cauchy t-test and the newly introduced martingale estimation technique. This dissertation nds no evidence of predictability in stock returns. Moreover, using martingale estimation, the cause of the Forward Premium Anomaly may be more easily discerned.

Since most financial institutions use the Sharpe Ratio to evaluate the performance of mutual funds, the objective of most fund managers is to select the portfolio that can generate the highest Sharpe Ratio. Traditionally, they can revise the objective function of the Markowitz mean-variance portfolio model and resolve non-linear programming to obtain the maximum Sharpe Ratio portfolio. In the scenario with short sales allowed, this project will propose a closed-form solution for the optimal Sharpe Ratio portfolio by applying Cauchy-Schwarz maximization. This method without using a non-linear programming computer program is easier than traditional method to implement and can save computing time and costs. Furthermore, in the scenarios with short sales disallowed, we will use Kuhn-Tucker conditions to find the optimal Sharpe Ratio portfolio. On the other hand, an efficient frontier generated by Markowitz mean-variance portfolio model normally has higher risk higher return characteristic, which often causes dilemma for decision maker. This research applies generalized loss function to create a family of decision-aid performance measures called IRp which can well tradeoff return with risk. We compare IRp with Sharpe Ratio and utility functions to confirm that IRp measures are approapriate to evaluate portfolio performance on efficient frontier and to improve asset allocation decisions. In addition, empirical data of domestic and international investment instruments will be used to examine the feasibility and fitness of the new proposed method and IRp measures. This study applies the methods of Cauchy-Schwarz maximization in multivariate statistical analysis and loss function in quality engineering to portfolio decisions. We believe these new applications will complete portfolio model theory and will be meaningful for academic and business fields.