On Lévy processes conditioned to stay positive.
Loïc Chaumont, LPMA - Université Paris 6
Ronald Arthur Doney, Department of Mathematics- University of Manchester
Abstract
We construct the law
of Lévy processes conditioned to stay positive under general
hypotheses. We obtain a Williams type path decomposition at the
minimum of these processes. This result is then applied to prove
the weak convergence of the law of Lévy processes conditioned to
stay positive as their initial state tends to 0. We describe an
absolute continuity relationship between the limit law and the
measure of the excursions away from 0 of the underlying Lévy
process reflected at its minimum. Then, when the Lévy process
creeps upwards, we study the lower tail at 0 of the law of the
height of this excursion.
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