A characterisation of, and hypothesis test for, continuous local martingales. (English) Zbl 1274.60140

Summary: 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 M. T. Barlow and E. A. Perkins [Probab. Theory Relat. Fields 79, No.4, 543-623 (1988; Zbl 0635.60090)]. 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.


60G44 Martingales with continuous parameter
62G10 Nonparametric hypothesis testing


Zbl 0635.60090
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