Spot estimation for fractional Ornstein–Uhlenbeck stochastic volatility model: consistency and central limit theorem

There has been an increasing interest for rough stochastic volatility models. However, little is known about the statistical inference for such models, especially for high frequency data. This paper investigates estimation of the fractional spot volatility from discrete observations of the price process on a grid with a time interval Δn→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _n\rightarrow 0$$\end{document} as n→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document}. Namely, the model with fractional Ornstein–Uhlenbeck log-volatility and Itô-semimartingale log-price processes is considered. In this setup both consistency and central limit theorem are proven for truncated and non-truncated spot volatility estimators. Then, asymptotic confidence intervals are derived for a finite number of spot volatility estimators at different estimation times. Consequently, the highest possible rate of convergence achieved in the central limit theorem ΔnH/(2H+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _n^{H/(2H+1)}$$\end{document} is a function of the Hurst parameter H of the fractional Brownian motion driving the volatility. This rate coincides with the already known highest convergence rate for the Brownian case when H=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=0.5$$\end{document}. Furthermore, simulations in this paper validate the consistency and central limit theorem numerically. Article class.