# Enlargements of filtrations and path decompositions at non stopping times

Nikeghbali, A (2006). Enlargements of filtrations and path decompositions at non stopping times. Probability Theory and Related Fields, 136(4):524-540.

## Abstract

Azéma associated with an honest time L the supermartingale $Z_{t}^{L}=\mathbb{P}[L>t|\mathcal{F}_{t}]$ and established some of its important properties. This supermartingale plays a central role in the general theory of stochastic processes and in particular in the theory of progressive enlargements of filtrations. In this paper, we shall give an additive characterization for these supermartingales, which in turn will naturally provide many examples of enlargements of filtrations. We combine this characterization with some arguments from both initial and progressive enlargements of filtrations to establish some path decomposition results, closely related to or reminiscent of Williams' path decomposition results. In particular, some of the fragments of the paths in our decompositions end or start with a new family of random times which are not stopping times, nor honest times.

## Abstract

Azéma associated with an honest time L the supermartingale $Z_{t}^{L}=\mathbb{P}[L>t|\mathcal{F}_{t}]$ and established some of its important properties. This supermartingale plays a central role in the general theory of stochastic processes and in particular in the theory of progressive enlargements of filtrations. In this paper, we shall give an additive characterization for these supermartingales, which in turn will naturally provide many examples of enlargements of filtrations. We combine this characterization with some arguments from both initial and progressive enlargements of filtrations to establish some path decomposition results, closely related to or reminiscent of Williams' path decomposition results. In particular, some of the fragments of the paths in our decompositions end or start with a new family of random times which are not stopping times, nor honest times.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Analysis Physical Sciences > Statistics and Probability Social Sciences & Humanities > Statistics, Probability and Uncertainty Progressive enlargements of filtrations - Initial enlargements of filtrations - Azéma's supermartingale - General theory of stochastic processes - Path decompositions - Pseudo-stopping times English 2006 20 Jan 2010 10:52 21 Jan 2022 14:23 Springer 0178-8051 The original publication is available at www.springerlink.com Green https://doi.org/10.1007/s00440-005-0493-9 http://arxiv.org/abs/math/0505623