@article{ECP1673,
author = {Owen Jones and David Rolls},
title = {A characterisation of, and hypothesis test for, continuous local martingales},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {16},
year = {2011},
keywords = {continuous martingale hypothesis; crossing-tree; realised volatility; time-change},
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 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.},
pages = {no. 56, 638-651},
issn = {1083-589X},
doi = {10.1214/ECP.v16-1673},
url = {http://ecp.ejpecp.org/article/view/1673}}