Projections, Pseudo-Stopping Times and the Immersion Property

Given two filtrations $\mathbb{F}\subset \mathbb{G}$, we study under which conditions the $\mathbb{F}$-optional projection and the $\mathbb{F}$-dual optional projection coincide for the class of $\mathbb{G}$-optional processes with integrable variation. It turns out that this property is equivalent to the immersion property for $\mathbb{F}$ and $\mathbb{G}$, that is every $\mathbb F$-localmartingaleisa$\mathbb G$-local martingale, which, equivalently, may be characterised using the class of $\mathbb{F}$-pseudo-stopping times. We also show that every $\mathbb{G}$-stopping time can be decomposed into the minimum of two barrier hitting times.