Volatility estimation in fractional Ornstein-Uhlenbeck models

TL;DR

This paper investigates the asymptotic behavior of the realized quadratic variation in fractional Ornstein-Uhlenbeck models driven by fractional Brownian motion, establishing convergence results and constructing a consistent volatility estimator.

Contribution

It provides new theoretical results on the convergence of quadratic variation and introduces a strongly consistent estimator for integrated volatility in fractional OU models.

Findings

Almost sure uniform convergence of quadratic variation

Stable weak convergence results

Construction of a consistent volatility estimator

Abstract

In this article we study the asymptotic behaviour of the realized quadratic variation of a process $\int_{0}^{t}u_{s}dY_{s}^{(1)}$% , where $u$ is a $\beta$-H\"older continuous process with $\beta > 1-H$ and $Y_{t}^{(1)}=\int_{0}^{t}e^{-s}dB^{H}_{a_s}$, where $a_{t}=He^{\frac{t% }{H}} $ and $B^H$ is a fractional Brownian motion, is connected to the fractional Ornstein-Uhlenbeck process of the second kind. We prove almost sure convergence uniformly in time, and a stable weak convergence for the realized quadratic variation. As an application, we construct strongly consistent estimator for the integrated volatility parameter in a model driven by $Y^{(1)}$.