We consider the estimation problem of the drift coefficient within the classes of diffusion processes. The Maximum Likelihood Estimator (MLE) is analyzed with regard to its pathwise stability properties and robustness towards misspecification in volatility and even the very nature of noise. We show that in dimension larger than one the classical MLE suffers from stability problems. To resolve this issue we construct a version of the estimator based on rough integrals (in the sense of T. Lyons) and present strong evidence that this construction resolves a number of stability issues inherent to the standard MLEs. We will also discuss some numerical examples to demonstrate the relevance of our results for applications.