Volatility Estimation of General Gaussian Ornstein-Uhlenbeck Process

TL;DR

This paper investigates the asymptotic properties of the realized quadratic variation of a Gaussian Ornstein-Uhlenbeck process driven by a self-similar Gaussian process, providing convergence results and a consistent volatility estimator.

Contribution

It establishes almost sure and stable weak convergence of the quadratic variation and introduces a new strongly consistent estimator for the integrated volatility.

Findings

Proved uniform almost sure convergence of quadratic variation.

Established stable weak convergence for the quadratic variation.

Constructed a strongly consistent volatility estimator.

Abstract

In this article we study the asymptotic behaviour of the realized quadratic variation of a process $\int_{0}^{t}u_{s}dG^{H}_{s}$, where $u$ is a $\beta$-H\"older continuous process with $\beta >1-H$ and $G^H$ is a self-similar Gaussian process with parameters $H\in(0,3/4)$. 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 $G^H$.