[Paper Review] Parameter estimation for fractional Ornstein-Uhlenbeck processes of general Hurst parameter

Yaozhong Hu, David Nualart+1 more|arXiv (Cornell University)|Mar 28, 2017

Stochastic processes and financial applications4 citations

TL;DR

This paper develops statistical estimators for the drift and volatility parameters of a fractional Ornstein-Uhlenbeck process driven by fractional Brownian motion with general Hurst parameter $H \in (0,1)$. It introduces higher-order power variations to estimate volatility and establishes almost sure convergence and central or noncentral limit theorems for the least squares estimator of the drift parameter across all $H$ ranges, fully resolving an open problem from prior work.

ABSTRACT

This paper provides several statistical estimators for the drift and volatility parameters of an Ornstein-Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time instants. First and higher order power variations are used to estimate the volatility parameter. The almost sure convergence of the estimators and the corresponding central limit theorems are obtained for all the Hurst parameter range $H\in (0, 1)$. The least squares estimator is used for the drift parameter. A central limit theorem is proved when the Hurst parameter $H \in (0, 1/2)$ and a noncentral limit theorem is proved for $H\in[3/4, 1)$. Thus, the open problem left in the paper by Hu and Nualart (2010) is completely solved, where a central limit theorem for least squares estimator is proved for $H\in [1/2, 3/4)$.

Motivation & Objective

• To develop consistent and asymptotically normal estimators for the drift and volatility parameters of a fractional Ornstein-Uhlenbeck process driven by fractional Brownian motion.
• To remove the prior restriction $H \leq 3/4$ in power variation-based estimation of integrated volatility.
• To establish central and noncentral limit theorems for the least squares estimator of the drift parameter $\theta$ across the full range $H \in (0,1)$, resolving an open problem from previous work.
• To provide a unified statistical framework for parameter estimation under both continuous and discrete observation schemes.

Proposed method

• Uses higher-order power variations $V^{n}_{k,p}(X)_T = \sum_{i=1}^{[nT]-k+1} \left| \sum_{j=0}^k (-1)^{k-j} \binom{k}{j} X_{(i+j-1)/n} \right|^p$ to estimate the integrated volatility $\int_0^T |\sigma_s|^p ds$.
• Applies the fourth moment theorem and central limit theorems for multiple Wiener-Ito chaos integrals to derive asymptotic normality of the volatility estimators.
• Employs the least squares estimator $\hat{\theta}_T = - \frac{\int_0^T X_t dX_t}{\int_0^T X_t^2 dt}$, where the stochastic integral is interpreted in the Skorohod sense.
• Establishes almost sure convergence of the volatility estimator via uniform convergence in probability over compact time intervals.
• Uses the representation $X_t = \sigma \int_0^t e^{-\theta(t-s)} dB_s^H + e^{-\theta t} \xi$ with $\xi = \sigma \int_{-\infty}^0 e^{\theta s} dB_s^H$ to analyze the stationary solution and apply ergodic theorems.
• Applies Pickands' theorem on the covariance structure of $Y_t$ to prove almost sure convergence of $Y_T / T^\alpha \to 0$ for $\alpha > 0$, supporting the ergodicity argument.

Experimental results

Research questions

• RQ1 Can higher-order power variations be used to consistently estimate the integrated volatility of a fractional Ornstein-Uhlenbeck process for all $H \in (0,1)$, beyond the prior restriction $H \leq 3/4$?
• RQ2 What is the asymptotic distribution of the least squares estimator for the drift parameter $\theta$ when $H \in (0,1/2)$?
• RQ3 How does the asymptotic behavior of the least squares estimator change when $H \in [3/4,1)$, and can a noncentral limit theorem be established?
• RQ4 Is the almost sure convergence of the volatility estimator valid uniformly in time and in probability over compact intervals?

Key findings

• The higher-order power variation estimator $|\hat{\sigma}_T|^p = \frac{n^{-1+pH} V^{n}_{k,p}(X)_T}{c_{k,p} T}$ converges almost surely to $|\sigma|^p$ as $n \to \infty$, uniformly in probability over compact time intervals.
• A central limit theorem is established for the volatility estimator when $H \in (0,1)$, with asymptotic variance depending on $c_{k,p}$ and the fourth cumulant of the Rosenblatt distribution.
• For the least squares estimator $\hat{\theta}_T$, a central limit theorem holds when $H \in (0,1/2)$, with $\sqrt{T}(\hat{\theta}_T - \theta) \xrightarrow{d} N(0, \sigma^2 \theta^{-1} / (2H-1)(4H-3))$.
• For $H \in [3/4,1)$, the limit distribution of $\sqrt{T}(\hat{\theta}_T - \theta)$ is non-Gaussian and converges to a Rosenblatt random variable, with explicit variance expression involving $H$ and $\theta$.
• The ergodic theorem implies $\frac{1}{T} \int_0^T X_t^2 dt \to \sigma^2 \theta^{-2H} H \Gamma(2H)$ almost surely and in $L^2$ as $T \to \infty$, for all $H \in (0,1)$.
• The almost sure convergence of $Y_T / T^\alpha \to 0$ for $\alpha > 0$ is proven for all $H \in (0,1)$, supporting the ergodicity and stationarity of the process $Y_t$.