A characterisation of, and hypothesis test for, continuous local martingales

Owen D. Jones (Dept. of Mathematics and Statistics, University of Melbourne)
David A. Rolls (Dept. of Psychological Sciences, University of Melbourne)


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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Pages: 638-651

Publication Date: October 21, 2011

DOI: 10.1214/ECP.v16-1673


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