Goodness-of-fit testing for fractional diffusions

This paper presents a goodness-of-fit test for the volatility function of a SDE driven by a Gaussian process with stationary and centered increments. Under rather weak assumptions on the Gaussian process, we provide a procedure for testing whether the unknown volatility function lies in a given linear functional space or not. This testing problem is highly non-trivial, because the volatility function is not identifiable in our model. The underlying fractional diffusion is assumed to be observed at high frequency on a fixed time interval and the test statistic is based on weighted power variations. Our test statistic is consistent against any fixed alternative.