## 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

### Keywords:

Ocone martinalges; invariance properties
Full Text: