Goodness of fit test for ergodic diffusions by tick time sample scheme

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 construct a test based on the data observed discretely in space, that is, the so-called tick time sampled data. It is proved that the asymptotic distribution of our test under the null hypothesis is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. It is also shown that the test is consistent under any fixed alternative.