Martingale Representation in Progressively Enlarged Lévy Filtrations

Abstract

In this paper we obtain a martingale representation theorem in the progressive enlargement $\mathbb{G}$ by a random time $\tau$ of the filtration ${\mathbb{F}}^L$ generated by a Lévy process $L$. The assumptions on the random time are that ${\mathbb{F}}^L$ is immersed in $\mathbb{G}$ and that $\tau$ avoids ${\mathbb{F}}^L$ stopping times. We also study the multiplicity of a progressively enlarged filtration.