Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

L'objectif de ce travail est d'étudier le comportement en temps long d'un modèle de particules avec une interaction de type branchement. Plus précisément, les particules se déplacent indépendamment suivant une dynamique markovienne jusqu'au temps de branchement, où elles donnent naissance à de nouvelles particules dont la position dépend de celle de leur mère et de son nombre d'enfants. Dans la première partie de ce mémoire nous omettons le branchement et nous étudions le comportement d'une seule lignée. Celle-ci est modélisée via un processus de Markov qui peut admettre des sauts, des parties diffusives ou déterministes par morceaux. Nous quantifions la convergence de ce processus hybride à l'aide de la courbure de Wasserstein, aussi nommée courbure grossière de Ricci. Cette notion de courbure, introduite récemment par Joulin, Ollivier, et Sammer correspond mieux à l'étude des processus avec sauts. Nous établissons une expression du gradient du semigroupe des processus de Markov stochastiquement monotone, qui nous permet d'expliciter facilement leur courbure. D'autres bornes fines de convergence en distance de Wasserstein et en variation totale sont aussi établies. Dans le même contexte, nous démontrons qu'un processus de Markov, qui change de dynamique suivant un processus discret, converge rapidement vers un équilibre, lorsque la moyenne des courbures des dynamiques sous-jacentes est strictement positive. Dans la deuxième partie de ce mémoire, nous étudions le comportement de toute la population de particules. Celui-ci se déduit du comportement d'une seule lignée grâce à une formule many-to-one, c'est-à-dire un changement de mesure de type Girsanov. Via cette transformation, nous démontrons une loi des grands nombres et établissons une limite macroscopique, pour comparer nos résultats aux résultats déjà connus en théorie des équations aux dérivées partielles. Nos résultats sont appliqués sur divers modèles ayant des applications en biologie et en informatique. Parmi ces modèles, nous étudierons le comportement en temps long de la plus grande particule dans un modèle simple de population structurée en taille / The aim of this work is to study the long time behavior of a branching particle model. More precisely, the particles move independently from each other following a Markov dynamics until the branching event. When one of these events occurs, the particle produces some random number of individuals whose position depends on the position of its mother and her number of offspring. In the first part of this thesis, we only study one particle line and we ignore the branching mechanism. So we are interested by the study of a Markov process which can jump, diffuse or be piecewise deterministic. The long time behavior of these hybrid processes is described with the notion of Wasserstein or coarse Ricci curvature. This notion of curvature, introduced by Joulin, Ollivier and Sammer, is more appropriate for the study of processes with jumps. We establish an expression of the gradient of the Markov semigroup of stochastically monotone processes which gives the curvature of these processes. Others sharp bounds of convergence, in Wasserstein distance and total variation distance, are also established. In the same way, we prove that if a Markov process evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of a second Markov process, then it is exponentially ergodic, under the assumption that the mean of the curvature of the underlying dynamics is positive. In the second part of the work, we study all the population. Its behaviour can be deduced to the study of the first part using a Girsavov-type transform which is called a many-to-one formula. Using this relation, we establish a law of large numbers and a macroscopic limit, in order to compare our results to the well know results on deterministic setting. Several examples, based on biology and computer science problems, illustrate our results, including the study of the largest individual in a size-structured population model

Dans cette thèse nous considérons le problème de la maximisation d’utilité et de la formation des prix d’indifférence pour les modèles semimartingales exponentiels dépendant d’un facteur aléatoire ξ. L’enjeu est de résoudre le problème des prix d’indifférence en utilisant le grossissement de l’espace et de la filtration. Nous réduisons le problème de maximisation dans la filtration élargie au problème conditionnel, sachant {ξ = v}, que nous résolvons en utilisant une approche duale. Pour HARA-utilités nous introduisons les informations telles que les entropies relatives et les intégrales de type Hellinger, ainsi que les processus d’information correspondants, enfin d’exprimer, via ces processus, l’utilité maximal. En particulier, nous étudions les modèles de Lévy exponentiels, où les processus d’information sont déterministes ce que simplifie considèrablement les calculs des prix d’indiffrence. Enfin, nous appliquons les rèsultats au modèle du mouvement brownien géométrique et au modèle de diffusion-sauts qui inclut le mouvement brownien et les processus de Poisson. Dans les cas d’utilité logarithmique, de puissance et exponentielle, nous fournissons les formules explicites des informations, et puis, en utilisant les méthodes numériques, nous résolvons les équations pour obtenir les prix d’indifférence en cas de vente d’une option européenne. / This thesis explores the utility maximisation problem and indifference pricing for exponential semimartingale models depending on a random factor ξ. The main idea to solve indifference pricing problem consists in the enlargement of the space and filtration. We reduce the maximization problem on the enlarged filtration to the conditional one, given {ξ = v}, which we solve using dual approach. For HARA-utilities we introduce the information quantities such that the relative entropies, Hellinger type integrals, and the corresponding information processes, and we express the maximal utility via these processes. As a particular case, we study exponential Levy models, where the information processes are deterministic and this fact simplify very much indifference price calculus. Finally, we apply the results to Geometric Brownian motion model and jump-diffusion model which incorporates Brownian motion and Poisson process. In the cases of logarithmic, power and exponential utilities, we provide the explicit formulae of information quantities and using the numerical methods we solve the equations for the seller’s and buyer’s indifference prices of European put option.

Ce manuscrit se concentre sur deux problèmes d'estimation de fonction. Pour chacun, une garantie non asymptotique des performances de l'estimateur proposé est fournie par une inégalité d'oracle. Pour l'estimation de densité conditionnelle, des mélanges de régressions gaussiennes à poids exponentiels dépendant de la covariable sont utilisés. Le principe de sélection de modèle par maximum de vraisemblance pénalisé est appliqué et une condition sur la pénalité est établie. Celle-ci est satisfaite pour une pénalité proportionnelle à la dimension du modèle. Cette procédure s'accompagne d'un algorithme mêlant EM et algorithme de Newton, éprouvé sur données synthétiques et réelles. Dans le cadre de la régression à bruit sous-gaussien, l'agrégation à poids exponentiels d'estimateurs linéaires permet d'obtenir une inégalité d'oracle en déviation, au moyen de techniques PAC-bayésiennes. Le principal avantage de l'estimateur proposé est d'être aisément calculable. De plus, la prise en compte de la norme infinie de la fonction de régression permet d'établir un continuum entre inégalité exacte et inexacte. / This manuscript focuses on two functional estimation problems. A non asymptotic guarantee of the proposed estimator’s performances is provided for each problem through an oracle inequality.In the conditional density estimation setting, mixtures of Gaussian regressions with exponential weights depending on the covariate are used. Model selection principle through penalized maximum likelihood estimation is applied and a condition on the penalty is derived. If the chosen penalty is proportional to the model dimension, then the condition is satisfied. This procedure is accompanied by an algorithm mixing EM and Newton algorithm, tested on synthetic and real data sets. In the regression with sub-Gaussian noise framework, aggregating linear estimators using exponential weights allows to obtain an oracle inequality in deviation,thanks to pac-bayesian technics. The main advantage of the proposed estimator is to be easily calculable. Furthermore, taking the infinity norm of the regression function into account allows to establish a continuum between sharp and weak oracle inequalities.

As doenças respiratórias e a poluição do ar são temas de muitos trabalhos científicos, porém a relação entre doenças respiratórias e queimadas de cana-de-açúcar ainda é pouco estudada. A queima da palha da cana-de-açúcar é uma prática comum em grande parte do Estado de São Paulo, com especial destaque para os dados da região de Ribeirão Preto. Os focos de queimadas são detectados por satélites do CPTEC/INPE (Centro de Previsão de Tempo e Estudos Climáticos do Instituto Nacional de Pesquisas Espaciais) e neste trabalho consideramos o tempo entre dias de extrapolação do número de internações diárias. Neste trabalho introduzimos diferentes modelos estatísticos para analisar dados de focos de queimadas e suas relações com as internações por doenças respiratórias. Propomos novos modelos para analisar estes dados, na presença ou não da covariável, que representa o número de queimadas. Sob o enfoque Bayesiano, usando os diferentes modelos propostos, encontramos os sumários a posteriori de interesse utilizando métodos de simulação de Monte Carlo em Cadeias de Markov. Também usamos técnicas Bayesianas para discriminar os diferentes modelos. Para os dados da região de Ribeirão Preto, encontramos modelos que levam à obtenção das inferências a posteriori com grande precisão e vericamos que a presença da covariável nos traz um grande ganho na qualidade dos dados ajustados. Os resultados a posteriori nos sugerem evidências de uma relação entre as queimadas e o tempo entre as extrapolações do número de internações, ou seja, de que quando observamos um maior número de queimadas anteriores à extrapolação, também observamos que o tempo entre as extrapolações é menor. / Relations between respiratory diseases and air pollution has been the goals of many scientic works, but the relation between respiratory diseases and sugar cane burning still is not well studied in the literature. Pre-harvest burning of sugarcane elds used primarily to get rid of the dried leaves is common in most of São Paulo state, Southeast Brazil, especially in the Ribeirão Preto region. The locals of pre-harvest sugar cane burning are detected by surveillance satellites of the CPTEC/INPE (Center of Climate Prediction of the Space Research National Institute). In this work, we consider as our data of interest, the time in days, between peaks numbers of hospitalizations due to respiratory diseases. Dierent statistical models are assumed to analyze the data of pre-harvest burning of sugar cane elds and their relations with hospitalizations due to respiratory diseases. These new models are considered to analyze data sets in presence or not of covariates, representing the numbers of pre-harvest burning of sugar cane elds. Under a Bayesian approach, we get the posterior summaries of interest using MCMC (Markov Chain Monte Carlo) methods. We also use dierent existing Bayesian discrimination methods to choose the best model. In our case, considering the data of Ribeirão Preto region, we observed that the models in presence of covariates give accurate inferences and good t for the data. We concluded that there is evidence of a relationship between respiratory diseases and sugar cane burning, that is, larger numbers of pre-harvest sugar cane burning, implies in larger numbers of hospitalizations due to respiratory diseases. In this case, we also observe small times (days) between extra numbers of hospitalizations.

Este trabalho é dedicado ao estudo do comportamento a longo prazo de uma equação de placas extensíveis acoplada a uma equação de calor do tipo hiperbólico. O problema corresponde a um modelo de termo-elasticidade baseado em teorias de calor do tipo não-Fourier. Considerando que efeitos de inércia de rotação estão presentes no modelo, mostramos que o efeito dissipativo do calor e suficiente para estabilizar exponencialmente o sistema, sem dissipações adicionais. Além disso, provamos que o sistema possui um atrator global de dimensão fractal finita e também atratores exponenciais. Nossos resultados generalizam e complementam diversos trabalhos existentes / This work is concerned with long-time dynamics of solutions of extensible plate equations with thermal memory. It corresponds to a model of thermoelasticity based on a theory of non-Fourier heat flux. By considering the case where rotational inertia is present we show that the thermal dissipation is sufficient to stabilize the system exponentially and guarantee the existence of a finite-dimensional global attractor. In addition the existence of an exponential attractor and some further properties are also considered. Our results complements several existing results

A técnica de análise de sobrevivência tem sido muito utilizada por pesquisadores na área de saúde. Neste trabalho foi usada uma distribuição em análise de sobrevivência recentemente estudada, chamada distribuição exponencial generalizada. Esta distribuição foi estudada sob todos os aspectos: para dados completos e censurados, sob a presençaa de covariáveis e considerando sua extensão para um modelo multivariado derivado de uma função cópula. Para exemplificação desta nova distribuição, foram utilizados dados reais de câncer (leucemia mielóide aguda e câncer gástrico) que possuem a presença de censuras e covariáveis. Os dados referentes ao câncer gástrico tem a particularidade de apresentar dois tempos de sobrevida, um relativo ao tempo global de sobrevida e o outro relativo ao tempo de sobrevida livre do evento, que foi utilizado para a aplicação do modelo multivariado. Foi realizada uma comparação com outras distribuições já utilizadas em análise de sobrevivência, como a distribuiçãoo Weibull e a Gama. Para a análise bayesiana adotamos diferentes distribuições a priori para os parâmetros. Foi utilizado, nas aplicações, métodos de simulação de MCMC (Monte Carlo em Cadeias de Markov) e o software Winbugs. / Survival analysis methods has been extensively used by health researchers. In this work it was proposed the use a survival analysis model recently studied, denoted as generalized exponential distribution. This distribution was studied in all respects: for complete data and censored, in the presence of covariates and considering its extension to a multivariate model derived from a copula function. To exemplify the use of these models, it was considered real cancer lifetime data (acute myeloid leukemia and gastric cancer) in presence of censored data and covariates. The assumed cancer gastric lifetime data has two survival responses, one related to the total lifetime of the patient and another one related to the time free of the disease, that is, multivariate data associated to each patient. In these applications there was considered a comparative study with standard existing lifetime distributions, as Weibull and gamma distributions.For a Bayesian analysis we assumed different prior distributions for the parameters of the model. For the simulation of samples of the joint posterior distribution of interest, we used standard MCMC (Markov Chain Monte Carlo) methods and the software Winbugs.

Der verhaltensökonomischen Literatur entsprechend behandeln die drei Kapitel dieser Dissertation unterschiedliche Aspekte des menschlichen Verhaltens, welches als "nicht-rational" zu bezeichnen ist. Jedes dieser Kapitel leistet einen Beitrag zum aktuellen Stand der Forschung auf dem Gebiet der Verhaltensökonomik mit Hilfe von entweder experimentellen, empirischen oder methodischen Ansätzen. Das erste Kapitel schlägt ein einfaches verhaltensökonomisches Modell vor und unterzieht dieses einer Reihe von experimentellen Tests. Das Modell erweitert die Literatur zur Fehlwahrnehmung von multiplikativen Wachstumsprozessen und hilft somit typische Fehlinvestitionen in der langen Frist zu erklären. Im Rahmen des zweiten Kapitels werden Daten einer Online-Kreditbörse genutzt, um empirisch zu untersuchen, ob sich private Investoren entsprechend den Vorhersagen der standardmäßigen ökonomischen Fachliteratur verhalten und einzig die erwartete Rendite berücksichtigen oder ob sie von anderen nicht-finanztechnischen Attributen eines Schuldners beeinflusst werden. Der Schwerpunkt der Analyse liegt dabei auf Geschlechterdiskriminierung im Rahmen dessen unterschiedliche Diskriminierungskonzepte getestet werden. Das dritte Kapitel wählt einen methodischen Ansatz und schlägt ein innovatives Experiment-Design vor, welches den empirisch gut dokumentierten Schwierigkeiten bzgl. der Angabe von subjektiven Wahrscheinlichkeiten von Teilnehmern an Umfragen und Laborexperimenten Rechnung trägt. Ein Binary-Choice-Ansatz eingebettet in ein adaptives Experiment-Design minimiert den Aufwand für die Befragten und ermöglich somit eine praktikable und effiziente Elizitierung der subjektiven Meinungen. / In the line with the literature on behavioral economics, the three chapters of this dissertation shed light on different aspects of human behavior that are at odds with rationality. Each chapter contributes to the existing behavioral economic research using either experimental, empirical, or methodological tools. First, by proposing and experimentally testing a simple behavioral model that extends the literature on the misperception of multiplicative growth processes, Chapter 1 aims to explain common money mistakes that people often make with long-term investments such as retirement savings plans. Second, in Chapter 2, real-life investment data of an online-lending platform are used to empirically investigate if private investors behave as the standard economic literature would predict and solely consider an investment’s expected return or if they also care about other non-financial attributes of a debtor. The focus of the analysis is on gender discrimination, thereby defining and econometrically testing different concepts of how investors discriminate between male and female borrowers. Third, Chapter 3 takes a methodological path and proposes a novel experimental design that accounts for the empirically well-documented difficulties that survey respondents typically have when asked to state subjective probabilities. A binary choice approach embedded in an adaptive experimental design helps to minimize effort of the respondents, thus allowing for a more practical belief elicitation in both the lab and the field.

In der Arbeit befassen wir uns mit der Potenznutzenmaximierung des Endvermögens, wenn die Aktienpreise stetigen Semimartingaldynamiken genügen und die Strategien des Agenten Investitions- und Informationsrestriktionen unterworfen sind. Hauptaugenmerk liegt auf der stochastischen Rückwärtsgleichung (BSDE) für den dynamischen Wertprozess und auf der Übertragung von neuen Ergebnissen zu quadratischen Semimartingal-BSDEs auf das Investitionsproblem. Dieses gelingt unter der Annahme endlicher exponentiellen Momente des Mean-Variance Tradeoff und verallgemeinert frühere Resultate, die Beschränktheit fordern. Wir betrachten dabei zunächst die Beziehung zwischen den Dualitäts- und BSDE-Ansätzen zur Lösung des Problems und gehen dann über zum Studium der quadratischen Semimartingal-BSDE, wenn der Marktpreis des Risikos vom BMO-Typ ist. Wir zeigen, dass es stets ein Kontinuum verschiedener BSDE-Lösungen mit quadratisch integrierbarem Martingalteil gibt. Wir stellen dann eine neue scharfe Bedingung an geeignete dynamische exponentielle Momente vor, die die Beschränktheit der BSDE-Lösungen in einer allgemeinen Filtration garantiert. In weiterer Folge weisen wir Existenz-, Eindeutigkeits-, Stabilitäts- und Maßwechselresultate für allgemeine quadratische stetige BSDEs unter exponentiellen Momenten nach. Diese Ergebnisse verwenden wir, um das Investitionsproblem für den Fall konischer Investitionsrestriktionen zu untersuchen. Ausgehend von der Zerlegung von Elementen des dualen Gebietes erhalten wir die zugehörige BSDE und beweisen, dass der Wertprozess in einem Raum liegt, in dem Lösungen quadratischer BSDEs eindeutig sind. Als Folgerung aus dem Stabilitätsresultat für BSDEs erhalten wir die Stetigkeit der Optimierer in der Semimartingaltopologie in den Parametern des Modells. Schließlich betrachten wir das Investitionsproblem unter exponentiellen Momenten, kompakten Handelsrestriktionen und eingeschränkter Information. Hierbei benutzen wir ausschließlich BSDE-Resultate. / In this thesis we consider the problem of maximizing the power utility from terminal wealth when the stocks have continuous semimartingale dynamics and there are investment and information constraints on the agent''s strategies. The main focus is on the backward stochastic differential equation (BSDE) that encodes the dynamic value process and on transferring new results on quadratic semimartingale BSDEs to the portfolio choice problem. This is accomplished under the assumption of finite exponential moments of the mean-variance tradeoff, generalizing previous results which require boundedness. We first recall the relationship between the duality and BSDE approaches to solving the problem and then study the associated quadratic semimartingale when the market price of risk is of BMO type. We show that there is always a continuum of distinct solutions to this BSDE with square-integrable martingale part. We then provide a new sharp condition on the dynamic exponential moments of the mean-variance tradeoff which guarantees the boundedness of BSDE solutions in a general filtration. In a subsequent step we establish existence, uniqueness, stability and measure change results for general quadratic continuous BSDEs under an exponential moments condition. We use these results to study the portfolio selection problem when there are conic investment constraints. Building on a decomposition result for the elements of the so-called dual domain we derive the associated BSDE and show that the value process is contained in a specific space in which BSDE solutions are unique. A consequence of the stability result for BSDEs is then the continuity of the optimizers with respect to the input parameters of the model in the semimartingale topology. Finally, we study the optimal investment problem under exponential moments, compact constraints and restricted information. This is done by referring to BSDE results only.

Os processos de decisão de Markov (Markov Decision Process - MDP) têm sido usados para resolução de problemas de tomada de decisão sequencial. Existem problemas em que lidar com os riscos do ambiente para obter um resultado confiável é mais importante do que maximizar o retorno médio esperado. MDPs que lidam com esse tipo de problemas são chamados de processos de decisão de Markov sensíveis a risco (Risk-Sensitive Markov Decision Process - RSMDP). Dentre as diversas variações de RSMDP, estão os trabalhos baseados em utilidade exponencial que utilizam um fator de risco, o qual modela a atitude a risco do agente e que pode ser propensa ou aversa. Os algoritmos existentes na literatura para resolver esse tipo de RSMDPs são ineficientes se comparados a outros algoritmos de MDP. Neste projeto, é apresentada uma solução que pode ser usada em problemas maiores, tanto por executar cálculos apenas em estados relevantes para atingir um conjunto de estados meta partindo de um estado inicial, quanto por permitir processamento de números com expoentes muito elevados para os ambientes computacionais atuais. Os experimentos realizados evidenciam que (i) o algoritmo proposto é mais eficiente, se comparado aos algoritmos estado-da-arte para RSMDPs; e (ii) o uso da técnica LogSumExp permite resolver o problema de trabalhar com expoentes muito elevados em RSMDPs. / Markov Decision Process (MDP) has been used very efficiently to solve sequential decision-making problems. There are problems where dealing with environmental risks to get a reliable result is more important than maximizing the expected average return. MDPs that deal with this type of problem are called risk-sensitive Markov decision processes (RSMDP). Among the several variations of RSMDP are the works based on exponential utility that use a risk factor, which models the agent\'s risk attitude that can be prone or averse. The algorithms in the literature to solve this type of RSMDPs are inefficient when compared to other MDP algorithms. In this project, a solution is presented that can be used in larger problems, either by performing calculations only in relevant states to reach a set of meta states starting from an initial state, or by allowing the processing of numbers with very high exponents for the current computational environments. The experiments show that (i) the proposed algorithm is more efficient when compared to state-of-the-art algorithms for RSMDPs; and (ii) the LogSumExp technique solves the problem of working with very large exponents in RSMDPs