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ELECTRONIC COMMUNICATIONS in PROBABILITY A CHARACTERISATION OF, AND HYPOTHESIS TEST FOR, CON- TINUOUS LOCAL MARTINGALES

By Owen D. Jones and David A. Rolls

Abstract

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 & 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.

Topics: continuous martingale hypothesis, crossing-tree, realised volatility, time-change
Year: 2010
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.3679
Provided by: CiteSeerX
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