  In this paper, we present a test for the maximal rank of the volatility
process in continuous diffusion models observed with noise. Such models are
typically applied in mathematical finance, where latent price processes are
corrupted by microstructure noise at ultra high frequencies. Using high
frequency observations we construct a test statistic for the maximal rank of
the time varying stochastic volatility process. Our methodology is based upon a
combination of a matrix perturbation approach and pre-averaging. We will show
the asymptotic mixed normality of the test statistic and obtain a consistent
testing procedure.
