  We present statistical tests for the continuous martingale hypothesis. That
is, whether an observed process is a continuous local martingale, or
equivalently a continuous time-changed Brownian motion. Our technique is based
on the concept of the crossing tree. Simulation experiments are used to assess
the power of the tests, which is generally higher than recently proposed tests
using the estimated quadratic variation (i.e., realised volatility). In
particular, the crossing tree shows significantly more power with shorter
datasets. We then show results from applying the methodology to high frequency
currency exchange rate data. We show that in 2003, for the AUD-USD, GBP-USD,
JPY-USD and EUR-USD rates, at small timescales (less than 15 minutes or so) the
continuous martingale hypothesis is rejected, but not so at larger timescales.
For 2003 EUR-GBP data, the hypothesis is rejected at small timescales and some
moderate timescales, but not all.
