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2011 A characterisation of, and hypothesis test for, continuous local martingales
Owen Jones, David Rolls
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Electron. Commun. Probab. 16: 638-651 (2011). DOI: 10.1214/ECP.v16-1673


We give characterisations for Brownian motion and continuous local martingales, using the crossing tree, which is a sample-path decomposition based on first-passages at nested scales. These results are based on ideas used in the construction of Brownian motion on the Sierpinski gasket (Barlow and Perkins 1988). Using our characterisation we propose a test for the continuous martingale hypothesis, that is, that a given process is a continuous local martingale. The crossing tree gives a natural break-down of a sample path at different spatial scales, which we use to investigate the scale at which a process looks like a continuous local martingale. Simulation experiments indicate that our test is more powerful than an alternative approach which uses the sample quadratic variation.


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Owen Jones. David Rolls. "A characterisation of, and hypothesis test for, continuous local martingales." Electron. Commun. Probab. 16 638 - 651, 2011.


Accepted: 21 October 2011; Published: 2011
First available in Project Euclid: 7 June 2016

zbMATH: 1274.60140
MathSciNet: MR2846657
Digital Object Identifier: 10.1214/ECP.v16-1673

Primary: 60G44
Secondary: 62G10


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