On Asymptotic Distribution of Parameter Free tests for Ergodic Diffusion Processes

We consider two problems of constructing of goodness of fit tests for ergodic diffusion processes. The first one is concerned with a composite basic hypothesis for a parametric class of diffusion processes, which includes the Ornstein-Uhlenbeck and simple switching processes. In this case we propose asymptotically parameter free tests of Cram\'er-von Mises type. The basic hypothesis in the second problem is simple and we propose asymptotically distribution free tests for a wider class of trend coefficients.