Pathwise stability of likelihood estimators for diffusions via rough paths

We consider the classical estimation problem of an unknown drift parameter within classes of nondegenerate diffusion processes. Using rough path theory (in the sense of T. Lyons), we analyze the Maximum Likelihood Estimator (MLE) with regard to its pathwise stability properties as well as robustness toward misspecification in volatility and even the very nature of the noise. Two numerical examples demonstrate the practical relevance of our results.