Bridges of Lévy processes conditioned to stay positive

Details Bridges of Lévy processes conditioned to stay positive Bridges of Lévy processes conditioned to stay positive

2011-01-21

arxiv.org/abs/1101.4184v2

Mathematics

Probability

12 pages; new introduction; results unchanged

Scientific paper

We consider Kallenberg's hypothesis on the characteristic function of a L\'evy process and show that it allows the construction of weakly continuous bridges of the L\'evy process conditioned to stay positive. We therefore provide a notion of normalized excursions of L\'evy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the L\'evy process conditioned to stay positive by using the weakly continuous bridges of the L\'evy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper semi-continuous versions. Using the bridges of the conditioned L\'evy process, the Durrett-Iglehart-Theorem stating that the Brownian bridge from 0 to 0 conditioned to remain above $-\eps$ converges weakly to the Brownian excursion as $\eps\to 0$, is extended to L\'evy processes. We also extend the Denisov decomposition of Brownian motion to L\'evy processes and their bridges, as well as Vervaat's classical result stating the equivalence in law of the Vervaat transform of a Brownian bridge and the normalized Brownian excursion.

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