Goodness-of-Fit Tests for Perturbed Dynamical Systems

We consider the goodness of fit testing problem for stochastic differential equation with small diffiusion coefficient. The basic hypothesis is always simple and it is described by the known trend coefficient. We propose several tests of the type of Cramer-von Mises, Kolmogorov-Smirnov and Chi-Square. The power functions of these tests we study for a special classes of close alternatives. We discuss the construction of the goodness of fit test based on the local time and the possibility of the construction of asymptotically distribution free tests in the case of composite basic hypothesis.