Quasi-stationary distributions and particular methods for the approximation of conditioned processes

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.