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Simplifiez vos Lévy en titillant la factorisation de Wierner-Hopf

Abstract : This thesis is devoted to the fluctuation theory of Lévy processes, discipline which studies trajectories, from the point of view of their local and global extrema. The main tool for this, is the \wh\ factorisation, which relates the exponent of the Lévy process with the exponents of the two famous ladder processes (the first one describes maxima, the second one describes minima). We ``titille" the \wh\ factorisation using its inverse Fourier transform and its analytic extention. This allows us to re-proove several cassical results (theorems of Rogozin , Bertoin, Kesten-Erickson, and the strong law of large numbers) with a simple analytic method. This method lead us to a criterion for creeping, formalted with the Lévy measure. It allows to recognize Lévy processes which, with positive probability, could cross continuously each positive altitude. This result answers a question that has been open for about 30 years. We also obtain a criterion for creeping formulated with marginal distributions, a criterion for the existence of increase points for creeping processes, and a condition for two Lévy exponents to appear in a \wh\ factorisation. A study of the bivariate ladder process provides informations on the suprema process $S_t=\sup\{ X_s : s\leq t\}$, where $X$ is the Lévy process under consideration. We show that, assuming the finiteness of an exponential moment, the distribution of $S$ characterizes that of $X$. When $X$ has unbounded variations, we see that the inferior limite of $\frac{S_t}{t}$, when $t$ tends to $0$ or $+\infty$, is equal to $0$ or $+\infty$. Finally, we characterize the cases where $S$ is piece-wise continuous. In the last part of this thesis, we study the relief of trajectories. We call abrupt processes the Lévy processes which have infinite right and left derivatives at their extrema. We give a characterisation of such processes and we analyse the Dini derivatives along their trajectories.
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Submitted on : Monday, February 21, 2011 - 12:03:38 PM
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  • HAL Id : tel-00567466, version 1



Vincent Vigon. Simplifiez vos Lévy en titillant la factorisation de Wierner-Hopf. Mathématiques [math]. INSA de Rouen, 2002. Français. ⟨NNT : 2002ISAM0002⟩. ⟨tel-00567466⟩



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