  We propose two nonparametric tests for investigating the pathwise properties
of a signal modeled as the sum of a L\'{e}vy process and a Brownian
semimartingale. Using a nonparametric threshold estimator for the continuous
component of the quadratic variation, we design a test for the presence of a
continuous martingale component in the process and a test for establishing
whether the jumps have finite or infinite variation, based on observations on a
discrete-time grid. We evaluate the performance of our tests using simulations
of various stochastic models and use the tests to investigate the fine
structure of the DM/USD exchange rate fluctuations and SPX futures prices. In
both cases, our tests reveal the presence of a non-zero Brownian component and
a finite variation jump component.
