RATE-OPTIMAL ESTIMATION OF MIXED SEMIMARTINGALES

• 成果类型: Article
• 署名作者: Chong, Carsten H.; Delerue, Thomas; Mies, Fabian
• 署名单位: Hong Kong University of Science & Technology; Delft University of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2461
• 发表日期: 2025
• 页码: 219-244
• 关键词: fractional gaussian-noise; ASYMPTOTIC THEORY; INTEGRATED VOLATILITY; microstructure noise; parameter; motion; memory

摘要

Consider the sum Y = B + B(H) of a Brownian motion B and an independent fractional Brownian motion B(H) with Hurst parameter H is an element of (0, 1). Even though B(H) is not a semimartingale, it was shown by Cheridito (Bernoulli 7 (2001) 913-934) that Y is a semimartingale if H > 3/4. Moreover, Y is locally equivalent to B in this case, so H cannot be consistently estimated from local observations of Y. This paper pivots on another unexpected feature in this model: if B and B(H) become correlated, then Y will never be a semimartingale, and H can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called mixed semimartingales, which are semiparametric extensions of Y with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense.