Review on goodness of fit tests for ergodic diffusion processes by different sampling schemes

We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional ergodic diffusions, where the diffusion coefficient is a nuisance function which is estimated in some sense. Using a theory for the continuous observation case, we first consider a test based on discrete time observations of the processes.Then we also construct a test based on the data observed discretely in space, that is, the so-called tick time sampled data. We prove that in both sampling schemes the limit distribution of the test is the supremum of the standard Brownian motion, thus the test is asymptotically distribution free. We also show that the tests are consistent under any fixed alternatives.