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Carthaginian enlargement of filtrations. (English) Zbl 1296.60106

Let \(\mathbb F\) be a filtration and \(\tau\) a finite random time whose \(\mathbb F\)-conditional law is equivalent to the law of \(\tau\) (“hypothesis \((\mathcal E)\)”). Let \(\mathbb G\) and \(\mathbb G^\tau\) denote the progressive and initial enlargements of \(\mathbb F\) with respect to \(\tau\), so that \(\mathbb F\subset\mathbb G\subset\mathbb G^\tau\). The title simply refers to these three levels of information, in analogy to the three levels of civilizations found at the archeological site of Carthage.
The paper studies various relations between martingales in these three filtrations. For martingales in the larger filtrations, it proves characterizations in terms of \(\mathbb F\)-martingales as well as predictable representation theorems. For \(\mathbb F\)-martingales viewed as semimartingales in the larger filtrations, it investigates their canonical decomposition. The main contribution is to provide an alternative approach to such results under hypothesis \((\mathcal E)\), which is slightly stronger than some previously used assumptions.

MSC:

60G44 Martingales with continuous parameter
60G40 Stopping times; optimal stopping problems; gambling theory
60H05 Stochastic integrals
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