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.
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