Abstract
We consider the initial and progressive enlargements of a Brownian filtration with a random time, that is, a strictly positive random variable. We assume Jacod’s equivalence hypothesis, that is, the existence of a strictly positive conditional density for the random time with respect to the Brownian filtration. Then, starting with the predictable integral representation of a martingale in the initially enlarged Brownian filtration, we derive explicit expressions for the components which appear in the predictable integral representations for the optional projections of the martingale on the progressively enlarged filtration and on the Brownian filtration. We also provide similar results for the optional projection of a martingale in the progressively enlarged filtration on the Brownian filtration.
Funding Statement
This research benefited from the support of the ’Chaire Marchés en Mutation’, French Banking Federation and ILB, Labex ANR 11-LABX-0019.
Acknowledgments
The authors thank warmly the anonymous referee for reading very carefully the two versions of the paper, pointing out many mistakes, misprints in the submitted versions. Following her/his questions and comments, and taking care about new references that she/he provided, we improved the quality of the paper. Monique Jeanblanc thanks Claudio Fontana and Marie-Claire Quenez for fruitful discussions.
Citation
Pavel V. Gapeev. Monique Jeanblanc. Dongli Wu. "Projections of martingales in enlargements of Brownian filtrations under Jacod’s equivalence hypothesis." Electron. J. Probab. 26 1 - 24, 2021. https://doi.org/10.1214/21-EJP694
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