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

A. Nikeghbali·M. Yor
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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.

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A. Nikeghbali, M. Yor. 2007-08-02. Doob's maximal identity, multiplicative decompositions and enlargements of filtrations. https://arxiv.org/abs/math/0503386

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