Projections, Pseudo-Stopping Times and the Immersion Property

TL;DR

This paper investigates the conditions under which optional projections coincide in filtered probability spaces, revealing their equivalence to the immersion property and characterizing pseudo-stopping times, with implications for stopping time decompositions.

Contribution

It establishes the equivalence between the projection coincidence property and the immersion property, and characterizes pseudo-stopping times within this framework.

Findings

Projection coincidence is equivalent to the immersion property.

Pseudo-stopping times characterize the projection property.

Any G-stopping time decomposes into two barrier hitting times.

Abstract

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.