Kollektive Migration und Schwarmverhalten sind Beispiele für Selbstorganisation und können in verschiedenen biologischen Systemen beobachtet werden, beispielsweise in Vogel-und Fischschwärmen oder Bakterienpopulationen. Im Zentrum dieser Arbeit steht ein räumlich diskretes und zeitlich stetiges Model, welches das kollektive Migrieren von Individuen mittels eines stochastischen Vielteilchensystems (VTS) beschreibt und analysierbar macht. Das konstruierte Modell ist in keiner Klasse gut untersuchter Vielteilchensysteme enthalten, sodass der größte Teil der Arbeit der Entwicklung von Methoden zur Untersuchung des Langzeitverhaltens bestimmter VTS gewidmet ist. Eine entscheidende Rolle spielen hier Gibbs-Maße, die zu zeitlich invarianten Maßen in Beziehung gesetzt werden. Durch eine Simulationsstudie und die Analyse des Einflusses der Parameter Migrationsgeschwindigkeit, Sensitivität der Individuen und (räumliche) Dichte der Anfangsverteilung können Eigenschaften kollektiver Migration erklärt und Hypothesen für weitere Analysen aufgestellt werden. / Collective migration and swarming behavior are examples of self-organization and can be observed in various biological systems, such as in flocks of birds, schools of fish or populations of bacteria. In the center of this thesis lies a stochastic interacting particle system (IPS), which is a spatially discrete model with a continuous time scale that describes collective migration and which can be treated using analytical methods. The constructed model is not contained in any class of well-understood IPS’s. The largest part of this work is used to develop methods that can be used to study the long-term behavior of certain IPS’s. Thereby Gibbs-Measures play an important role and are related to temporally invariant measures. One can explain the properties of collective migration and propose a hypothesis for further analyses by a simulation study and by analysing the parameters migration velocity, sensitivity of individuals and (spatial) density of the initial distribution.

Dans cette thèse, on étudie un système de particules en interaction qui généralise un processus de contact, évoluant en environnement aléatoire. Le processus de contact peut être interprété comme un modèle de propagation d'une population ou d'une infection. La motivation de ce modèle provient de la biologie évolutive et de l'écologie comportementale via la technique du mâle stérile, il s'agit de contrôler une population d'insectes en y introduisant des individus stérilisés de la même espèce: la progéniture d'une femelle et d'un individu stérile n'atteignant pas de maturité sexuelle, la population se voit réduite jusqu'à potentiellement s'éteindre. Pour comprendre ce phénomène, on construit un modèle stochastique spatial sur un réseau dans lequel la population suit un processus de contact dont le taux de croissance est ralenti en présence d'individus stériles, qui forment un environnement aléatoire dynamique. Une première partie de ce document explore la construction et les propriétés du processus sur le réseau Z^d. On obtient des conditions de monotonie afin d'étudier la survie ou la mort du processus. On exhibe l'existence et l'unicité d'une transition de phase en fonction du taux d'introduction des individus stériles. D'autre part, lorsque d=1 et cette fois en fixant l'environnement aléatoire initialement, on exhibe de nouvelles conditions de survie et de mort du processus qui permettent d'expliciter des bornes numériques pour la transition de phase. Une seconde partie concerne le comportement macroscopique du processus en étudiant sa limite hydrodynamique lorsque l'évolution microscopique est plus complexe. On ajoute aux naissances et aux morts des déplacements de particules. Dans un premier temps sur le tore de dimension d, on obtient à la limite un système d'équations de réaction-diffusion. Dans un second temps, on étudie le système en volume infini sur Z^d, et en volume fini, dans un cylindre dont le bord est en contact avec des réservoirs stochastiques de densités différentes. Ceci modélise des phénomènes migratoires avec l'extérieur du domaine que l'on superpose à l'évolution. À la limite on obtient un système d'équations de réaction-diffusion, auquel s'ajoutent des conditions de Dirichlet aux bords en présence de réservoirs. / In this thesis, we study an interacting particle system that generalizes a contact process, evolving in a random environment. The contact process can be interpreted as a spread of a population or an infection. The motivation of this model arises from behavioural ecology and evolutionary biology via the sterile insect technique ; its aim is to control a population by releasing sterile individuals of the same species: the progeny of a female and a sterile male does not reach sexual maturity, so that the population is reduced or potentially dies out. To understand this phenomenon, we construct a stochastic spatial model on a lattice in which the evolution of the population is governed by a contact process whose growth rate is slowed down in presence of sterile individuals, shaping a dynamic random environment. A first part of this document investigates the construction and the properties of the process on the lattice Z^d. One obtains monotonicity conditions in order to study the survival or the extinction of the process. We exhibit the existence and uniqueness of a phase transition with respect to the release rate. On the other hand, when d=1 and now fixing initially the random environment, we get further survival and extinction conditions which yield explicit numerical bounds on the phase transition. A second part concerns the macroscopic behaviour of the process by studying its hydrodynamic limit when the microscopic evolution is more intricate. We add movements of particles to births and deaths. First on the d-dimensional torus, we derive a system of reaction-diffusion equations as a limit. Then, we study the system in infinite volume in Z^d, and in a bounded cylinder whose boundaries are in contact with stochastic reservoirs at different densities. As a limit, we obtain a non-linear system, with additionally Dirichlet boundary conditions in bounded domain.

Kollektive Migration und Schwarmverhalten sind Beispiele für Selbstorganisation und können in verschiedenen biologischen Systemen beobachtet werden, beispielsweise in Vogel-und Fischschwärmen oder Bakterienpopulationen. Im Zentrum dieser Arbeit steht ein räumlich diskretes und zeitlich stetiges Model, welches das kollektive Migrieren von Individuen mittels eines stochastischen Vielteilchensystems (VTS) beschreibt und analysierbar macht. Das konstruierte Modell ist in keiner Klasse gut untersuchter Vielteilchensysteme enthalten, sodass der größte Teil der Arbeit der Entwicklung von Methoden zur Untersuchung des Langzeitverhaltens bestimmter VTS gewidmet ist. Eine entscheidende Rolle spielen hier Gibbs-Maße, die zu zeitlich invarianten Maßen in Beziehung gesetzt werden. Durch eine Simulationsstudie und die Analyse des Einflusses der Parameter Migrationsgeschwindigkeit, Sensitivität der Individuen und (räumliche) Dichte der Anfangsverteilung können Eigenschaften kollektiver Migration erklärt und Hypothesen für weitere Analysen aufgestellt werden. / Collective migration and swarming behavior are examples of self-organization and can be observed in various biological systems, such as in flocks of birds, schools of fish or populations of bacteria. In the center of this thesis lies a stochastic interacting particle system (IPS), which is a spatially discrete model with a continuous time scale that describes collective migration and which can be treated using analytical methods. The constructed model is not contained in any class of well-understood IPS’s. The largest part of this work is used to develop methods that can be used to study the long-term behavior of certain IPS’s. Thereby Gibbs-Measures play an important role and are related to temporally invariant measures. One can explain the properties of collective migration and propose a hypothesis for further analyses by a simulation study and by analysing the parameters migration velocity, sensitivity of individuals and (spatial) density of the initial distribution.

Cette thèse est consacrée à l’analyse de ces modèles particulaires pour les mathématiques financières.Le manuscrit est organisé en quatre chapitres. Chacun peut être lu séparément.Le premier chapitre présente le travail de thèse de manière globale, définit les objectifs et résume les principales contributions. Le deuxième chapitre constitue une introduction générale à la théorie des méthodes particulaire, et propose un aperçu de ses applications aux mathématiques financières. Nous passons en revue les techniques et les résultats principaux sur les systèmes de particules en interaction, et nous expliquons comment ils peuvent être appliques à la solution numérique d’une grande variété d’applications financières, telles que l’évaluation d’options compliquées qui dépendent des trajectoires, le calcul de sensibilités, l’évaluation d’options américaines ou la résolution numérique de problèmes de contrôle et d’estimation avec observation partielle.L’évaluation d’options américaines repose sur la résolution d’une équation d’évolution à rebours, nommée l’enveloppe de Snell dans la théorie du contrôle stochastique et de l’arrêt optimal. Les deuxième et troisième chapitres se concentrent sur l’analyse de l’enveloppe de Snell et de ses extensions à différents cas particuliers. Un ensemble de modèles particulaires est alors proposé et analysé numériquement. / This thesis is concerned with the analysis of these particle models for computational finance.The manuscript is organized in four chapters. Each of them could be read separately.The first chapter provides an overview of the thesis, outlines the motivation and summarizes the major contributions. The second chapter gives a general in- troduction to the theory of interacting particle methods, with an overview of their applications to computational finance. We survey the main techniques and results on interacting particle systems and explain how they can be applied to the numerical solution of a variety of financial applications; to name a few: pricing complex path dependent European options, computing sensitivities, pricing American options, as well as numerically solving partially observed control and estimation problems.The pricing of American options relies on solving a backward evolution equation, termed Snell envelope in stochastic control and optimal stopping theory. The third and fourth chapters focus on the analysis of the Snell envelope and its variation to several particular cases. Different type of particle models are proposed and studied.

Um conjunto de modelos compostos de redes de filas em grades finitas servindo como ambientes aleatorios para um ou mais passeios aleatorios, que por sua vez podem afetar o comportamento das filas, e desenvolvido. Duas formas de interacao entre os passeios aleatorios sao consideradas. Para cada modelo, e provado que o processo Markoviano correspondente e recorrente positivo e reversivel. As equacoes de balanceamento detalhado sao analisadas para obter a forma funcional da medida invariante de cada modelo. Em todos os modelos analisados neste trabalho, a medida invariante em uma grade finita tem forma produto. Modelos de redes de filas como ambientes para multiplos passeios aleatorios sao estendidos a grades infinitas. Para cada modelo estendido, sao especificadas as condicoes para a existencia do processo estocastico na grade infinita. Alem disso, e provado que existe uma unica medida invariante na rede infinita cuja projecao em uma subgrade finita e dada pela medida correspondente de uma rede finita. Finalmente, e provado que essa medida invariante na rede infinita e reversivel. / A set of models composed of queueing networks serving as random environments for one or more random walks, which themselves can affect the behavior of the queues, is developed. Two forms of interaction between the random walkers are considered. For each model, it is proved that the corresponding Markov process is positive recurrent and reversible. The detailed balance equa- tions are analyzed to obtain the functional form of the invariant measure of each model. In all the models analyzed in the present work, the invariant measure on a finite lattice has product form. Models of queueing networks as environments for multiple random walks are extended to infinite lattices. For each model extended, the conditions for the existence of the stochastic process on the infinite lattice are specified. In addition, it is proved that there exists a unique invariant measure on the infinite network whose projection on a finite sublattice is given by the corresponding finite- network measure. Finally, it is proved that that invariant measure on the infinite lattice is reversible.

Um conjunto de modelos compostos de redes de filas em grades finitas servindo como ambientes aleatorios para um ou mais passeios aleatorios, que por sua vez podem afetar o comportamento das filas, e desenvolvido. Duas formas de interacao entre os passeios aleatorios sao consideradas. Para cada modelo, e provado que o processo Markoviano correspondente e recorrente positivo e reversivel. As equacoes de balanceamento detalhado sao analisadas para obter a forma funcional da medida invariante de cada modelo. Em todos os modelos analisados neste trabalho, a medida invariante em uma grade finita tem forma produto. Modelos de redes de filas como ambientes para multiplos passeios aleatorios sao estendidos a grades infinitas. Para cada modelo estendido, sao especificadas as condicoes para a existencia do processo estocastico na grade infinita. Alem disso, e provado que existe uma unica medida invariante na rede infinita cuja projecao em uma subgrade finita e dada pela medida correspondente de uma rede finita. Finalmente, e provado que essa medida invariante na rede infinita e reversivel. / A set of models composed of queueing networks serving as random environments for one or more random walks, which themselves can affect the behavior of the queues, is developed. Two forms of interaction between the random walkers are considered. For each model, it is proved that the corresponding Markov process is positive recurrent and reversible. The detailed balance equa- tions are analyzed to obtain the functional form of the invariant measure of each model. In all the models analyzed in the present work, the invariant measure on a finite lattice has product form. Models of queueing networks as environments for multiple random walks are extended to infinite lattices. For each model extended, the conditions for the existence of the stochastic process on the infinite lattice are specified. In addition, it is proved that there exists a unique invariant measure on the infinite network whose projection on a finite sublattice is given by the corresponding finite- network measure. Finally, it is proved that that invariant measure on the infinite lattice is reversible.