Bridges of Lévy processes conditioned to stay positive

Abstract

We consider Kallenberg’s hypothesis on the characteristic function of a Lévy process and show that it allows the construction of weakly continuous bridges of the Lévy process conditioned to stay positive. We therefore provide a notion of normalized excursions Lévy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the Lévy process conditioned to stay positive by using the weakly continuous bridges of the Lévy 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 Lévy process, the Durrett–Iglehart theorem stating that the Brownian bridge from $0$ to $0$ conditioned to remain above $-\varepsilon $ converges weakly to the Brownian excursion as $\varepsilon \to0$, is extended to Lévy processes. We also extend the Denisov decomposition of Brownian motion to Lévy 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.