Rate-optimal estimation of mixed semimartingales

Journal article

Carsten Chong, Thomas Delerue, Fabian Mies
arXiv:2207.10464, 2022



APA   Click to copy
Chong, C., Delerue, T., & Mies, F. (2022). Rate-optimal estimation of mixed semimartingales. ArXiv:2207.10464.

Chicago/Turabian   Click to copy
Chong, Carsten, Thomas Delerue, and Fabian Mies. “Rate-Optimal Estimation of Mixed Semimartingales.” arXiv:2207.10464 (2022).

MLA   Click to copy
Chong, Carsten, et al. “Rate-Optimal Estimation of Mixed Semimartingales.” ArXiv:2207.10464, 2022.

BibTeX   Click to copy

  title = {Rate-optimal estimation of mixed semimartingales},
  year = {2022},
  journal = {arXiv:2207.10464},
  author = {Chong, Carsten and Delerue, Thomas and Mies, Fabian}

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). Surprisingly, even though B(H) is not a semimartingale, Cheridito proved in [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 a second surprise 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. The estimation of mixed semimartingales with correlation is motivated by applications to high-frequency financial data contaminated by rough noise.

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