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Chong, C.H.* ; Delerue, T. ; Mies, F.*

Rate-optimal estimation of mixed semimartingales.

Ann. Stat. 53, 219-244 (2025)
DOI
Open Access Green as soon as Postprint is submitted to ZB.
Consider the sum Y = B + B(H) of a Brownian motion B and an independent fractional Brownian motion B(H) with Hurst parameter H ∈ (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.
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Publication type Article: Journal article
Document type Scientific Article
Keywords Central Limit Theorem ; High-frequency Observations ; Hurst Parameter ; Kl Divergence ; Minimax Rate ; Mixed Fractional Brownian Motion ; Rough Noise; Fractional Gaussian-noise; Asymptotic Theory; Integrated Volatility; Microstructure Noise; Parameter; Motion; Memory
ISSN (print) / ISBN 0090-5364
Journal Annals of Statistics, The
Quellenangaben Volume: 53, Issue: 1, Pages: 219-244 Article Number: , Supplement: ,
Publisher Institute of Mathematical Statistics (IMS)
Publishing Place 3163 Somerset Dr, Cleveland, Oh 44122 Usa
Reviewing status Peer reviewed
Institute(s) Institute of Epidemiology (EPI)
Grants
DFG
ECS project