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

We review some recent results on goodness of fit test for the drift coefficient of a one-dimensional ergodic diffusion, where the diffusion coefficient is a nuisance function which however is estimated. Using a theory for the continuous observation case, we first present a test based on deterministic discrete time observations of the process. Then we also propose a test based on the data observed discretely in space, that is, the so-called tick time sample scheme. 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. The tests are also consistent under any fixed alternatives.