In this PhD thesis, we address the problem of giving simultaneous confidence statements about local features of the diffusion of an Itô process. To this end, we construct a multiscale test based on weighted quadratic variation and prove that the test statistic can be strongly approximated by a sequence of Gaussian martingales which are distribution-free. Further, we give optimality results and present different visualization methods.
In the second part of the thesis, we extend the approach to data corrupted by additive noise to cover applications from high-frequency finance. Additionally, we show which difficulties arise from real data and apply our method exemplarily to prices of Euro-Bund-Futures (FGBL).
As an outlook for future work, we present ideas of generalizing the method to inference on the local covariance and point out some interesting applications from finance.
