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

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

2005-03-18

arxiv.org/abs/math/0503386v2

Illinois Journal of Mathematics, 50 (4), 791-814 (2006)

Mathematics

Probability

Typos corrected

Scientific paper

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.

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