Before revision:  Our setup encompasses a large class of Gaussian processes and we show how to extend it to a large class of non-Gaussian models as well. It is proved that the asymptotic distribution of the estimator of \alpha does not depend on the specifics of the data generating process for the observations, but only on the value of \alpha and a "heteroskedasticity" factor. Using this, we propose a simulation-based approach to inference, which is easily implemented and is valid more generally than asymptotic analysis. We detail how the methods can be applied to test whether a stochastic process is a non-semimartingale .
After revision:  Our setup encompasses a large class of Gaussian processes and we show how to extend it to a large class of non-Gaussian processes. Particular focus is on clarity and ease of implementation of the estimator and the associated asymptotic results, making it easy for practitioners to apply the methods. We further develop a new estimator which is robust to measurement noise in the observations .