Doob's maximal identity, multiplicative decompositions and enlargements of filtrations

Abstract

In the theory of progressive enlargements of filtrations, the supermartingale $Z_{t}=\mathbf{P}( g>t\mid \mathcal{F}_{t}) $ associated with an honest time $g$, and its additive (Doob-Meyer) decomposition, play an essential role. In this paper, we propose an alternative approach, using a multiplicative representation for the supermartingale $Z_{t}$, based on Doob's maximal identity. We thus give new examples of progressive enlargements. Moreover, we give, in our setting, a proof of the decomposition formula for martingales , using initial enlargement techniques, and use it to obtain some path decompositions given the maximum or minimum of some processes.