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 Electronic Journal of Probability > Vol. 2 (1997) > Paper 6 open journal systems 


Cauchy's Principal Value of Local Times of Lévy Processes with no Negative Jumps via Continuous Branching Processes

Jean Bertoin, Université Paris VI


Abstract
Let $X$ be a recurrent Lévy process with no negative jumps and $n$ the measure of its excursions away from $0$. Using Lamperti's connection that links $X$ to a continuous state branching process, we determine the joint distribution under $n$ of the variables
$C^+_T=int_{0}^{T}{bf 1}_{{X_s>0}}X_s^{-1}ds$ and
$C^-_T=int_{0}^{T}{bf 1}_{{X_s<0}}|X_s|^{-1}ds$, where $T$ denotes the duration of the excursion. This provides a new insight on an identity of Fitzsimmons and Getoor on the Hilbert transform of the local times of $X$. Further results in the same vein are also discussed.


Full text: PDF

Pages: 1-12

Published on: September 1, 1997
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Electronic Journal of Probability. ISSN: 1083-6489