Immersion of strongly Brownian filtrations with honest time avoiding stopping times

Abstract

In this paper, we give a partial answer to the following question: if $\mathbb{F}\hookrightarrow\mathbb{G}\hookrightarrow\mathbb{H}$ (where the symbol ($\hookrightarrow$) indicates the immersion property), $\mathbb{F}$ and $\mathbb{H}$ are two strongly Brownian filtrations, is $\mathbb{G}$ also a strongly Brownian filtration ?\\We prove that $\mathbb{G}$ is weakly Brownian in the case of progressive enlargement of $\mathbb{F}$ with an honest time $\tau$ that avoids all stopping times.