Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - University of Rouen Normandy

The aim of this work is to develop a testing procedure which determines the rank of the noise in a two-dimensional stochastic process from discrete observations of this process on a fixed time interval $[0,T]$ sampled with a fixed time step $\Delta$. First, we construct the main statistics of the test, given by a random matrix determinant, as proposed in Jacod et Podolskij (2013). We show that the performance of the test based on this statistics is limited in a non-asymptotic setting, when $\Delta$ is fixed. Then, we show how the performance  of the test can be improved by centering the increments of the process around their expected value, given by the drift term. Finally, we derive the distribution of the centered statistics and show under which conditions the Type I and Type II errors of the test can be  controlled.