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Summary. Consider a semimartingale of the form Yt = Y0 + t 0 asds + t 0 σs− dWs, where a is a locally bounded predictable process and σ (the "volatility") is an adapted right-continuous process with left limits and W is a Brownian motion. We consider the realised bipower variation processwhere r and s are nonnegative reals with r + s > 0. We prove that V (Y ; r, s) n t converges locally uniformly in time, in probability, to a limiting process V (Y ; r, s)t (the "bipower variation process"). If further σ is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with W and by a Poisson random measure, we prove that √ n (V (Y ; r, s) n − V (Y ; r, s)) converges in law to a process which is the stochastic integral with respect to some other Brownian motion W , which is independent of the driving terms of Y and σ. We also provide a multivariate version of these results, and a version in which the absolute powers are replaced by smooth enough functions.

We propose a new concept of modulated bipower variation for diffusion models with microstructure noise. We show that this method provides simple estimates for such important quantities as integrated volatility or integrated quarticity. Under mild conditions the consistency of modulated bipower variation is proven. Under further assumptions we prove stable convergence of our estimates with the optimal rate n −1/4 . Moreover, we construct estimates which are robust to finite activity jumps. 1 0 σ 4 s ds.This quantity is of particular interest because, properly scaled, it occurs as the conditional variance in central limit theorems for estimators of IV and has therefore to be estimated.

International audienceWe show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi-Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data