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Rate-optimal estimation of mixed semimartingales.
Ann. Stat. 53, 219-244 (2025)
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
ISSN (print) / ISBN
0090-5364
Journal
Annals of Statistics, The
Quellenangaben
Volume: 53,
Issue: 1,
Pages: 219-244
Publisher
Institute of Mathematical Statistics (IMS)
Non-patent literature
Publications
Reviewing status
Peer reviewed
Institute(s)
Institute of Epidemiology (EPI)