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Some invariance properties (of the laws) of Ocone’s martingales. (English) Zbl 0965.60047

Azéma, J. (ed.) et al., Séminaire de Probabilités XXXIV. Berlin: Springer. Lect. Notes Math. 1729, 417-431 (2000).
The authors introduce the notion of Ocone martingales as those martingales \(M\) whose Dubins-Schwarz representation as a time-changed Brownian motion \(M=\beta_{\langle M \rangle}\) features independent \(\beta\) and \(\langle M \rangle\). They provide a characterization of these martingales both in terms of Doléans-Dade exponentials and in terms of Girsanov transformations. It is shown that an Ocone martinagle enjoys the martingale representation property with respect to its natural filtration iff its bracket process \(\langle M \rangle\) is deterministic. Properties of martinagles with respect to the natural filtration of an Ocone martingale or its bracket process are derived. The Ocone martingales within a large class of stochastic integrals of Brownian motion are characterized. The paper closes with a number of examples of Ocone martingales and non-Ocone martingales.
For the entire collection see [Zbl 0940.00007].

MSC:

60G44 Martingales with continuous parameter
60G15 Gaussian processes
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