Assume that high-frequency data from a stochastic process X are recorded under additive measurement noise.

For various applications, the quantity of interest is the spot volatility of X which measures the local variability of the process. In this talk, we study nonparametric estimation of the spot volatility. Minimax rates and adaptive estimators are derived. We evaluate the performance of the estimators by Monte Carlo simulations. A small empirical study is presented based on high-frequency financial data and further applications to turbulence modeling and biology are discussed.