Integral representations of martingales for progressive enlargements of filtrations

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.