A large deviation principle for the normalized excursion of $α$-stable Lévy processes without negative jumps

Abstract: We establish a large deviation principle for the normalized excursion and bridge of an $\alpha$-stable L\'evy process without negative jumps, with $1<\alpha<2$. Based on this, we derive precise asymptotics for the tail distributions of functionals of the normalized excursion and bridge, in particular, the area and maximum functionals. We advocate the use of the Skorokhod M1 topology, rather than the more usual J1 topology, as we believe it is better suited to large deviation principles for L\'evy processes in general.