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


Nikeghbali, A; Yor, M (2006). Doob's maximal identity, multiplicative decompositions and enlargements of filtrations. Illinois Journal of Mathematics, 50(1-4):791-814 (electronic).

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.

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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2006
Deposited On:20 Jan 2010 11:18
Last Modified:05 Apr 2016 13:24
Publisher:University Of Illinois At Urbana-Champaign, Department of Mathematics
ISSN:0019-2082
ISBN:0-9746986-1-X
Official URL:http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ijm/1258059492
Related URLs:http://arxiv.org/abs/math/0503386

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Content: Accepted Version
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