Sur la théorie des excursions pour des processus de Lévy symétriques stables d'indice α ϵ ]1,2] et quelques applications

Abstract : This thesis consists of five chapters. Chapter 1 is divided in two parts; first part concerns the general Lévy processes and the second one, the particular case of symmetric stable Lévy processes. Chapter 2 focuses on the theory of fluctuations in the stable case and contains most of the original results of this thesis. In this chapter, we study the joint distribution of the first passage time over a fixed barrier and the corresponding overshoot. We then look into the joint distribution of the process at time t, its supremum before t and the last hitting time of the supremum before t. Chapter 3 is also composed of two parts. First part treats local times and the second deals with the excursion theory. Both are studied in the case of symmetric stable Lévy processes with index greater than 1. For the local times we give the fundamental definitions and classical properties. Concerning the excursion theory, we develop an exposition in a classical way. We provide, among others, the definitions of normalized excursion and meander. We also give simple constructions for these objects. Finally, some recent results due to K.Yano, Y. Yano and M. Yor are exposed. Chapters 4 and 5 deal with applications (in the symmetric stable case) of excursions theory. We study the time spent positive and negative. and generalised principal values.
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Fernando Cordero. Sur la théorie des excursions pour des processus de Lévy symétriques stables d'indice α ϵ ]1,2] et quelques applications. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2010. Français. ⟨tel-00521136⟩

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