Projections, Pseudo-Stopping Times and the Immersion Property

© Springer International Publishing Switzerland 2016. Given two filtrations F ⊂ G, we study under which conditions the Foptional projection and the F-dual optional projection coincide for the class of G-optional processes with integrable variation. It turns out that this property is equivalent to the immersion property for F and G, that is every F-local martingale is a G-local martingale, which, equivalently, may be characterised using the class of F-pseudo-stopping times. We also show that every G-stopping time can be decomposed into the minimum of two barrier hitting times.