Estimating time-changes in noisy Lévy models

In quantitative finance, we often model asset prices as a noisy It\^o semimartingale. As this model is not identifiable, approximating by a time-changed L\'evy process can be useful for generative modelling. We give a new estimate of the normalised volatility or time change in this model, which obtains minimax convergence rates, and is unaffected by infinite-variation jumps. In the semimartingale model, our estimate remains accurate for the normalised volatility, obtaining convergence rates as good as any previously implied in the literature.