The asymptotics of the integrated self-weighted cross volatility estimator

In this paper, we are concerned with the inference of the integrated self-weighted cross volatility, ∫01g(Xt,Yt)σtXYdt, where g is some real function, σtXY is the instantaneous cross volatility of two continuous semi-martingales X and Y. We assume that processes X and Y are sampled with microstructure noise and in an asynchronous way. The asymptotic normality is investigated and a consistent estimator of the resulting limiting conditional variance is presented yielding a studentized central limit theorem. Simulation is given to check the performance of the theory.