Estimating time-changes in noisy Lévy models

In quantitative finance, we often wish to model the behaviour of asset prices given by a noisy Ito semimartingale; unfortunately, this model is too complex to identify from price data. In this paper, we therefore consider efficient prices given by a time-changed Levy process; this model is both identifiable, and replicates salient features of price data. We give a new estimate of the rate process in this model, which governs its volatility. Our estimate obtains minimax convergence rates, and is unaffected by arbitrary jump activity. Furthermore, it remains valid for the volatility in the general semimartingale model, obtaining convergence rates as good as any previously implied in the literature.