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 $\mathbb F$-local martingale is a $\mathbb G$-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.