Volatility estimation in fractional Ornstein-Uhlenbeck models

Abstract

In this article we study the asymptotic behaviour of the realized quadratic variation of a process t 0 u s dY (1) s , where u is a β-Hölder continuous process with β > 1−H and Y (1) t = t 0 e −s dB H as , where a t = He 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) . t 0 u s dB H s with respect to a fractional Brownian motion was studied in [8]. In [8] the authors proved 1 that if u = {u t , t 0} has finite q-variation for some q < 1/(1 − H), then n −1+pH V n p (Z) t −→ c 1,p t 0

Berry-Esseen bounds of second moment estimators for Gaussian processes observed at high frequency

“…We have the following: Corollary 3.5. Assume there exists 0 < β < 3 4 such that, |ρ(t)| = O(t 2β−2 ). Then,…”

Volatility Estimation of General Gaussian Ornstein-Uhlenbeck Process

“…We begin with two simple propositions which allows us to introduce drift to the process defined by (15), which can be obtained directly from Bajja, Es-Sebaiy and Viitasaari [2], so we omit the detailed proof here. almost surely and uniformly in t.…”

“…To the best of our knowledge, stronger mode of convergence such as uniform almost sure convergence is not widely studied in the literature. In the paper [2], they studied the asymptotic behaviour of the realized quadratic variation of a process of the form…”