[Paper Review] Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind

Ehsan Azmoodeh, Lauri Viitasaari|arXiv (Cornell University)|Apr 9, 2013

Stochastic processes and financial applications12 references4 citations

TL;DR

This paper proposes consistent estimators for the drift parameter θ in the fractional Ornstein-Uhlenbeck process of the second kind (fOU₂) based on discrete-time observations. Using Malliavin calculus and ergodicity, it establishes central limit theorems for estimators when H is known and when H is unknown, valid for all H ∈ (1/2, 1), under appropriate sampling conditions on the observation window and mesh size.

ABSTRACT

Fractional Ornstein-Uhlenbeck process of the second kind $( ext{fOU}_{2})$ is solution of the Langevin equation $\mathrm{d}X_t = -θX_t\,\mathrm{d}t+\mathrm{d}Y_t^{(1)}, \ θ>0$ with Gaussian driving noise $ Y_t^{(1)} := \int^t_0 e^{-s} \,\mathrm{d}B_{a_s}$, where $ a_t= H e^{\frac{t}{H}}$ and $B$ is a fractional Brownian motion with Hurst parameter $H \in (0,1)$. In this article, we consider the case $H>\frac{1}{2}$. Then using the ergodicity of $ ext{fOU}_{2}$ process, we construct consistent estimators of drift parameter $θ$ based on discrete observations in two possible cases: $(i)$ the Hurst parameter $H$ is known and $(ii)$ the Hurst parameter $H$ is unknown. Moreover, using Malliavin calculus technique, we prove central limit theorems for our estimators which is valid for the whole range $H \in (\frac{1}{2},1)$.

Motivation & Objective

• To develop consistent estimators for the drift parameter θ in the fOU₂ process based on discrete-time observations.
• To establish central limit theorems (CLT) for the estimators under the assumption that the Hurst parameter H is known.
• To extend the CLT framework to the case where H is unknown, under restricted sampling conditions.
• To leverage the ergodicity of the fOU₂ process and Malliavin calculus techniques to derive asymptotic normality results.
• To provide a rigorous statistical inference framework for fOU₂ processes, which exhibit short memory regardless of H > 1/2.

Proposed method

• Utilizes the ergodicity of the fOU₂ process to construct estimators based on discrete sampling over a growing observation window T_N = NΔ_N.
• Applies Malliavin calculus techniques to analyze the asymptotic distribution of the estimators, particularly focusing on the Wiener chaos expansion and multiple Wiener integrals.
• Derives the asymptotic variance of the estimators by computing the limit of the second moment of the score function using change of variables and beta function identities.
• Imposes mesh conditions: T_N → ∞ and NΔ_N² → 0 when H is known; Δ_N = N^{-α} with α ∈ (1/2, 1/(4H−2) ∧ 1) when H is unknown.
• Reduces the covariance structure of the estimator to a triple integral involving exponential and power-law terms, which is shown to converge to a finite limit via change of variables and beta function bounds.
• Uses L’Hôpital’s rule and asymptotic analysis to compute the limit of the derivative of the variance expression, confirming convergence to a non-degenerate limit.

Experimental results

Research questions

• RQ1 Can consistent estimators for the drift parameter θ be constructed from discrete observations of the fOU₂ process when H is known?
• RQ2 What sampling conditions ensure asymptotic normality of the drift estimator when H is known?
• RQ3 Can a consistent and asymptotically normal estimator for θ be constructed when H is unknown?
• RQ4 What are the necessary and sufficient sampling conditions on the observation mesh for the CLT to hold in the unknown H case?
• RQ5 How does the use of Malliavin calculus enable the derivation of CLT results for fOU₂ processes with non-semimartingale noise?

Key findings

• A strongly consistent estimator for θ is constructed when H is known, under the conditions T_N → ∞ and NΔ_N² → 0.
• The asymptotic distribution of the estimator is normal with a non-degenerate variance that is explicitly characterized via a triple integral involving power-law and exponential terms.
• For the unknown H case, a consistent estimator is constructed under the mesh condition Δ_N = N^{-α} with α ∈ (1/2, 1/(4H−2) ∧ 1), ensuring asymptotic normality.
• The central limit theorem holds for the entire range H ∈ (1/2, 1), which is a significant improvement over previous results limited to H ∈ [1/2, 3/4).
• The asymptotic variance is shown to be finite by bounding the relevant triple integral using change of variables and beta function identities, confirming the existence of the limit.
• The second-order term in the covariance function is shown to vanish asymptotically, justifying the use of the simplified kernel 1/(2θ)e^{-θ|x−y|} in the limit analysis.