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Volatility estimation under one-sided errors with applications to limit order books. (English) Zbl 1353.60044

Summary: For a semi-martingale \(X_{t}\), which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation \(\langle X,X\rangle_{t}\) is constructed based on observations in the vicinity of \(X_{t}\). The problem is embedded in a Poisson point process framework, which reveals an interesting connection to the theory of Brownian excursion areas. We derive \(n^{-1/3}\) as optimal convergence rate in a high-frequency framework with \(n\) observations (in mean). We discuss a potential application for the estimation of the integrated squared volatility of an efficient price process \(X_{t}\) from intra-day order book quotes.

MSC:

60G48 Generalizations of martingales
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
60F99 Limit theorems in probability theory
60H30 Applications of stochastic analysis (to PDEs, etc.)
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