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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.

MSC:

60G44 Martingales with continuous parameter
62G10 Nonparametric hypothesis testing

Citations:

Zbl 0635.60090
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