Dans cette thèse, l'étude menée consiste à établir de nouveaux résultats théoriques concernant des problèmes d'existence et d'unicité pour des Equations Différentielles Stochastiques Rétrogrades (EDSR) à croissance quadratique : ceci a pour but de permettre la résolution d'un problème de Mathématiques Financières, à savoir la maximisation de l'utilité (exponentielle) d'un portefeuille sous contraintes. Généralisant des résultats déjà connus en filtration brownienne pour les EDSR quadratiques, ce travail permet ainsi d'apporter des réponses au problème financier dans des contextes plus généraux.

M.H.A. Davis. / Bibliography: leaves 30-33. / "January, 1979." / U.S. Air Force Office of Sponsored Research Grant AFOSR 77-3281 Department of Energy Contract EX-76-A-01-2295

This thesis deals with credit risk modeling and related mathematical issues. In particular we study first-passage models for credit risk, where obligors default upon first passage of a ``credit quality" process to zero. The first passage problem for correlated Brownian motion is a mathematical structure which arises quite naturally in such models, in particular the seminal multivariate Black-Cox model. In general this problem is analytically intractable, however in two dimensions analytic results are available. In addition to correcting mistakes in several published formulae, we derive an exact simulation scheme for sampling the passage times. Our algorithm exploits several interesting properties of planar Brownian motion and conformal local martingales. The main contribution of this thesis is the development of a novel multivariate framework for credit risk. We allow for both stochastic trend and volatility in credit qualities, with dependence introduced by letting these quantities be driven by systematic factors common to all obligors. Exploiting a conditional independence structure we are able to express the proportion of defaults in an asymptotically large portfolio as a path functional of the systematic factors. The functional in question returns crossing probabilities of time-changed Brownian motion to continuous barriers, and is typically not available in closed form. As such the distribution of portfolio losses is in general analytically intractable. As such we devise a scheme for simulating approximate losses and demonstrate almost sure convergence of this approximation. We show that the model calibrates well, across both tranches and maturities, to market quotes for CDX index tranches. In particular we are able to calibrate to data from 2006, as well as more recent ``distressed" data from 2008.

This thesis deals with credit risk modeling and related mathematical issues. In particular we study first-passage models for credit risk, where obligors default upon first passage of a ``credit quality" process to zero. The first passage problem for correlated Brownian motion is a mathematical structure which arises quite naturally in such models, in particular the seminal multivariate Black-Cox model. In general this problem is analytically intractable, however in two dimensions analytic results are available. In addition to correcting mistakes in several published formulae, we derive an exact simulation scheme for sampling the passage times. Our algorithm exploits several interesting properties of planar Brownian motion and conformal local martingales. The main contribution of this thesis is the development of a novel multivariate framework for credit risk. We allow for both stochastic trend and volatility in credit qualities, with dependence introduced by letting these quantities be driven by systematic factors common to all obligors. Exploiting a conditional independence structure we are able to express the proportion of defaults in an asymptotically large portfolio as a path functional of the systematic factors. The functional in question returns crossing probabilities of time-changed Brownian motion to continuous barriers, and is typically not available in closed form. As such the distribution of portfolio losses is in general analytically intractable. As such we devise a scheme for simulating approximate losses and demonstrate almost sure convergence of this approximation. We show that the model calibrates well, across both tranches and maturities, to market quotes for CDX index tranches. In particular we are able to calibrate to data from 2006, as well as more recent ``distressed" data from 2008.

Martingales are fundamental stochastic process used to model the concept of fair game. They have a multitude of applications in the real world that include, random walks, Brownian motion, gamblers fortunes and survival analysis, Just as commutative integration theory may be realised as a special case of the more general non-commutative theory for integrals, so too, we find classical probability may be realised as a limiting, special case of quantum probability theory. In this thesis we are concerned with the development of multiparameter quantum stochastic integrals extending non-commutative constructions to the general n parameter case, these being multiparameter quantum stochastic integrals over the positive n - dimensional plane, employing martingales as integrator. The thesis extends previous analogues of type one, and type two stochastic integrals, for both Clifford and quasi free representations. As with one and two dimensional parameter sets, the stochastic integrals constructed form orthogonal, centred L2 - martingales, obeying isometry properties. We further explore analogues for weakly adapted processes, properties relating to the resulting quantum stochastic integrals, develop analogues to Fubini’s theorem, and explore applications for quantum stochastic integrals in a security setting.

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (p. 118-120).

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: Joseph Silverman. Includes bibliographical references (leaves 87-90). Also available online.

[pt] Métodos da teoria de martingais tem sido amplamente utilizados em matemática nos últimos decênios. Mais recentemente, eles também vêm sendo usados em matemática atuarial. Nesta tese discutimos um exemplo de aplicação desta metodologia na demonstração do teorema clássico de Cramér-Lundberg para o problema da ruína. / [en] In the last decades martingale theory tools have been extensively in mathematical finance. More recently, they have also been used in actuarial mathematics. In this thesis we illustrate this methodology in the proof of the classical Lundberg-Crámer theorem for the ruin problem.

Cette thèse est consacrée à l'étude de certains modèles de déformation semi-paramétriques. Notre objectif est de proposer des méthodes récursives, issues d'algorithmes stochastiques, pour estimer les paramètres de ces modèles. Dans la première partie, on présente les outils théoriques existants qui nous seront utiles dans la deuxième partie. Dans un premier temps, on présente un panorama général sur les méthodes d'approximation stochastique, en se focalisant en particulier sur les algorithmes de Robbins-Monro et de Kiefer-Wolfowitz. Dans un second temps, on présente les méthodes à noyaux pour l'estimation de fonction de densité ou de régression. On s'intéresse plus particulièrement aux deux estimateurs à noyaux les plus courants qui sont l'estimateur de Parzen-Rosenblatt et l'estimateur de Nadaraya-Watson, en présentant les versions récursives de ces deux estimateurs.Dans la seconde partie, on présente tout d'abord une procédure d'estimation récursive semi-paramétrique du paramètre de translation et de la fonction de régression pour le modèle de translation dans la situation où la fonction de lien est périodique. On généralise ensuite ces techniques au modèle vectoriel de déformation à forme commune en estimant les paramètres de moyenne, de translation et d'échelle, ainsi que la fonction de régression. On s'intéresse finalement au modèle de déformation paramétrique de variables aléatoires dans le cadre où la déformation est connue à un paramètre réel près. Pour ces trois modèles, on établit la convergence presque sûre ainsi que la normalité asymptotique des estimateurs paramétriques et non paramétriques proposés. Enfin, on illustre numériquement le comportement de nos estimateurs sur des données simulées et des données réelles. / This thesis is devoted to the study of some semi-parametric deformation models.Our aim is to provide recursive methods, related to stochastic algorithms, in order to estimate the different parameters of the models. In the first part, we present the theoretical tools which we will use in the next part. On the one hand, we focus on stochastic approximation methods, in particular the Robbins-Monro algorithm and the Kiefer-Wolfowitz algorithm. On the other hand, we introduce kernel estimators in order to estimate a probability density function and a regression function. More particularly, we present the two most famous kernel estimators which are the one of Parzen-Rosenblatt and the one of Nadaraya-Watson. We also present their recursive version.In the second part, we present the results we obtained in this thesis.Firstly, we provide a recursive estimation method of the shift parameter and the regression function for the translation model in which the regression function is periodic. Secondly, we extend this estimation procedure to the shape invariant model, providing estimation of the height parameter, the translation parameter and the scale parameter, as well as the common shape function.Thirdly, we are interested in the parametric deformation model of random variables where the deformation is known and depending on an unknown parameter.For these three models, we establish the almost sure convergence and the asymptotic normality of each estimator. Finally, we numerically illustrate the asymptotic behaviour of our estimators on simulated data and on real data.

Cette thèse présente quelques résultats de la théorie des probabilités non commutatives, et traite en particulier des inégalités de martingales dans des algèbres de von Neumann et de leurs espaces de Hardy associés. La première partie démontre un analogue non commutatif de la décomposition de Davis faisant intervenir la fonction carrée. Les arguments classiques de temps d'arrêt ne sont plus valides dans ce cadre, et la preuve se base sur une approche duale. Le deuxième résultat important de cette partie détermine ainsi le dual de l'espace de Hardy conditionnel h_1(M). Ces résultats sont ensuite étendus au cas 1<p<2. La deuxième partie transfère une décomposition atomique pour les espaces de Hardy h_1(M) et H_1(M) aux martingales non commutatives. Des résultats d'interpolation entre les espaces h_p(M) et bmo(M) sont également établis, relativement aux méthodes complexe et réelle d'interpolation. Les deux premières parties concernent des filtrations discrètes. Dans la troisième partie, on introduit des espaces de Hardy de martingales non commutatives relativement à une filtration continue. Les analogues des inégalités de Burkholder/Gundy et de Burkholder/Rosenthal sont obtenues dans ce cadre. La dualité de Fefferman-Stein ainsi que la décomposition de Davis sont également transférées avec succès à cette situation. Les preuves se basent sur des techniques d'ultraproduit et de L_p-modules. Une discussion sur une décomposition impliquant des atomes algébriques permet d'obtenir les résultats d'interpolation attendus / This thesis presents some results of the theory of noncommutative probability. It deals in particular with martingale inequalities in von Neumann algebras, and their associated Hardy spaces. The first part proves a noncommutative analogue of the Davis decomposition, involving the square function. The usual arguments using stopping times in the commutative case are no longer valid in this setting, and the proof is based on a dual approach. The second main result of this part determines the dual of the conditioned Hardy space h_1(M). These results are then extended to the case 1<p<2. The second part proves that an atomic decomposition for the Hardy spaces h_1(M) and H_1(M) is valid for noncommutative martingales. Interpolation results between the spaces h_p(M) and bmo(M) are also established, with respect to both complex and real interpolations. The two first parts concern discrete filtrations. In the third part, we introduce Hardy spaces of noncommutative martingales with respect to a continuous filtration. The analogues of the Burkholder/Gundy and Burkholder/Rosenthal inequalities are obtained in this setting. The Fefferman-Stein duality and the Davis decomposition are also successfully transferred to this situation. The proofs are based on ultraproduct techniques and L_p-modules. A discussion about a decomposition involving algebraic atoms gives the expected interpolation results