Martingale Representation in Progressively Enlarged L\'evy Filtrations

TL;DR

This paper establishes a martingale representation theorem within a progressively enlarged filtration by a random time for a Lévy process, under specific immersion and avoidance assumptions, and explores the filtration's multiplicity.

Contribution

It provides a new martingale representation theorem in the context of Lévy processes with progressive enlargement by a random time, under certain immersion and avoidance conditions.

Findings

Martingale representation theorem established for Lévy processes with enlarged filtration.

Conditions include immersion of the original filtration and avoidance of stopping times by the random time.

Analysis of the multiplicity of the enlarged filtration.

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 L\'evy 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.