For a semi-martingale Xt, which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation (X;X)t is constructed based on observations in the vicinity of Xt. The problem is embedded in a Poisson point process framework, which reveals an interesting connection to the theory of Brownian excursion areas. A major application is the estimation of the integrated squared volatility of an effcient price process Xt from intra-day order book quotes. We derive n -1/3 as optimal convergence rate of integrated squared volatility estimation in a high-frequency framework with n observations (in mean). This considerably improves upon the classical n -1/4-rate obtained from transaction prices under microstructure noise.
Brownian excursion area limit order book integrated volatility Feynman-Kac high-frequency data Poisson point process