Goodness of fit test for small diffusions by discrete observations

We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional small diffusions. Our test is based on discrete observation of the processes, and the diffusion coefficient is a nuisance function which is estimated in our testing procedure. We prove that the limit distribution of our test is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. We also show that our test is consistent under any fixed alternatives