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Distributions quasi-stationnaires et méthodes particulaires pour l'approximation de processus conditionnés

Abstract : My PhD thesis focuses on the study of the distributions of stochastic processes with absorption and their approximation. This processes are commonly used in a large area of applications in ecology, finance or reliability studies. In particular, we study the long term evolution of the distribution of Markov processes with absorption. Non-trivial behaviors, like mortality plateaus, can be described and explained by the limiting distribution of a process conditioned not to be absorbed when it is observed. When such a limiting distribution exists, it is called a quasi-stationary distribution. In the first chapter, we recall and prove in all generality some specific properties of these distributions. In the following chapters, we prove in a great generality an approximation method based on particle systems in order to approximate the distribution of conditioned Markov processes and their quasi-stationary distributions. Programs written in C++ during my thesis allow us to present a numerical implementation of this approximation method for biological models, like the Wright-Fisher diffusion process or the Lotka-Volterra diffusion processes. The approximation method proved in this thesis associated with coupling technics allows us to obtain new results of existence and uniqueness of quasi-stationnary distributions. Moreover, we show some mixing properties for diffusion processes conditioned to remain in a bounded open subset.
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Submitted on : Monday, February 20, 2012 - 2:40:28 PM
Last modification on : Wednesday, March 27, 2019 - 4:08:31 PM

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Denis Villemonais. Distributions quasi-stationnaires et méthodes particulaires pour l'approximation de processus conditionnés. Probabilités [math.PR]. Ecole Polytechnique X, 2011. Français. ⟨pastel-00672074⟩

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