Integral representations of martingales for progressive enlargements of filtrations

TL;DR

This paper investigates integral representations of martingales in the context of progressive filtration enlargements, focusing on Poisson filtrations and pseudo-stopping times, and establishes predictable representation properties under these settings.

Contribution

It provides new integral representation results for ext{G}-martingales in progressive enlargements, extending known properties to broader classes of filtrations and random times.

Findings

Established predictable representation for Poisson filtrations.

Extended integral representations to pseudo-stopping times.

Identified conditions for representation with fewer martingales.

Abstract

We work in the setting of the progressive enlargement $\mathbb G$ of a reference filtration $\mathbb F$ through the observation of a random time $\tau$. We study an integral representation property for some classes of $\mathbb G$-martingales stopped at $\tau$. In the first part, we focus on the case where $\mathbb F$ is a Poisson filtration and we establish a predictable representation property with respect to three $\mathbb G$-martingales. In the second part, we relax the assumption that $\mathbb F$ is a Poisson filtration and we assume that $\tau$ is an $\mathbb F$-pseudo-stopping time. We establish integral representations with respect to some $\mathbb G$-martingales built from $\mathbb F$-martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two $\mathbb G$-martingales.