Bridges of L\'{e}vy processes conditioned to stay positive

TL;DR

This paper constructs continuous transition densities for Levy processes conditioned to stay positive, extending classical Brownian results to Levy processes using weakly continuous bridges and Hunt's method.

Contribution

It introduces a novel construction of continuous transition densities for Levy processes conditioned to stay positive, generalizing Brownian excursion theory.

Findings

Constructed weakly continuous bridges of Levy processes conditioned to stay positive

Extended the Durrett-Iglehart theorem to Levy processes

Generalized Vervaat's transformation and Denisov decomposition to Levy processes

Abstract

We consider Kallenberg's hypothesis on the characteristic function of a L\'{e}vy process and show that it allows the construction of weakly continuous bridges of the L\'{e}vy process conditioned to stay positive. We therefore provide a notion of normalized excursions L\'{e}vy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the L\'{e}vy process conditioned to stay positive by using the weakly continuous bridges of the L\'{e}vy 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\'{e}vy 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 L\'{e}vy processes. We also extend the Denisov decomposition of Brownian motion to L\'{e}vy 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.