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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Pages: 948-961

Publication Date: July 14, 2005

DOI: 10.1214/EJP.v10-261

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