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Drift estimation for a partially observed mixed fractional Ornstein--Uhlenbeck process

Chunhao Cai

TL;DR

This work develops drift estimation for a partially observed Ornstein--Uhlenbeck process driven by mixed fractional Brownian noise. By constructing the canonical innovation representation and applying Kalman filtering with a Riccati error-variance equation, the authors derive a four-dimensional linear system that supports the Laplace-transform analysis of the likelihood. Within the Ibragimov–Khasminskii LAN framework, they prove that the MLE is consistent and asymptotically normal at rate $\sqrt{T}$, with a closed-form Fisher information $\mathcal{I}(\vartheta)$ identical to the classical Brownian/Kalman–Bucy case, i.e., $${\mathcal I}(\vartheta)=\frac{1}{2\vartheta}-\frac{2\vartheta}{\alpha(\alpha+\vartheta)}+\frac{\vartheta^2}{2\alpha^3},\quad \alpha=\sqrt{\mu^2+\vartheta^2}.$$ The paper further analyzes the long-time behavior of the filtering covariance via a mixed-Laplace condition and performs a spectral analysis of the effective Hamiltonian to confirm the Fisher information and its independence from the Hurst parameter. Simulation results illustrate the asymptotic normality of the MLE and validate the theoretical variance. Overall, the study unifies partially observed OU drift estimation with mixed fractional noise, showing that the asymptotic information content remains unchanged from the classical Brownian setting.

Abstract

We consider estimation of the drift parameter $\vartheta>0$ in a \emph{partially observed} Ornstein--Uhlenbeck type model driven by a mixed fractional Brownian noise. Our framework extends the partially observed model of \cite{BrousteKleptsyna2010} to the \emph{mixed} case. We construct the canonical innovation representation, derive the associated Kalman filter and Riccati equations, and analyse the asymptotic behaviour of the filtering error covariance. Within the Ibragimov--Khasminskii LAN framework we prove that the MLE of $\vartheta$, based on continuous observation of the partially observed system on $[0,T]$, is consistent and asymptotically normal with rate $\sqrt{T}$ and the Fisher Information is the same as in \cite{BrousteKleptsyna2010} or the standard Brownian motion case.

Drift estimation for a partially observed mixed fractional Ornstein--Uhlenbeck process

TL;DR

This work develops drift estimation for a partially observed Ornstein--Uhlenbeck process driven by mixed fractional Brownian noise. By constructing the canonical innovation representation and applying Kalman filtering with a Riccati error-variance equation, the authors derive a four-dimensional linear system that supports the Laplace-transform analysis of the likelihood. Within the Ibragimov–Khasminskii LAN framework, they prove that the MLE is consistent and asymptotically normal at rate , with a closed-form Fisher information identical to the classical Brownian/Kalman–Bucy case, i.e., The paper further analyzes the long-time behavior of the filtering covariance via a mixed-Laplace condition and performs a spectral analysis of the effective Hamiltonian to confirm the Fisher information and its independence from the Hurst parameter. Simulation results illustrate the asymptotic normality of the MLE and validate the theoretical variance. Overall, the study unifies partially observed OU drift estimation with mixed fractional noise, showing that the asymptotic information content remains unchanged from the classical Brownian setting.

Abstract

We consider estimation of the drift parameter in a \emph{partially observed} Ornstein--Uhlenbeck type model driven by a mixed fractional Brownian noise. Our framework extends the partially observed model of \cite{BrousteKleptsyna2010} to the \emph{mixed} case. We construct the canonical innovation representation, derive the associated Kalman filter and Riccati equations, and analyse the asymptotic behaviour of the filtering error covariance. Within the Ibragimov--Khasminskii LAN framework we prove that the MLE of , based on continuous observation of the partially observed system on , is consistent and asymptotically normal with rate and the Fisher Information is the same as in \cite{BrousteKleptsyna2010} or the standard Brownian motion case.

Paper Structure

This paper contains 19 sections, 3 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Canonical representation of the mixed model
  3. Mixed fractional Brownian motion and the fundamental martingale
  4. State--observation model in innovation form
  5. Kalman filtering and error covariance
  6. Kalman filter
  7. Filtering error covariance and Riccati equation
  8. Laplace transform and four-dimensional system
  9. Likelihood ratio and Laplace transform
  10. Four-dimensional linear system
  11. Riccati equation and linearisation
  12. Asymptotics of the covariance and the Laplace condition
  13. Rescaled covariance and its limit
  14. Asymptotic form of the drift matrix
  15. Derivation of the Fisher Information via Spectral Analysis of the Effective Hamiltonian
  16. ...and 4 more sections

Key Result

Lemma 2.1

There exists $t_0>0$ such that $\psi\in C^1[t_0,\infty)$, $\psi(t)>0$ for all $t\ge t_0$, and Moreover, there exist constants $0<c_1\le c_2<\infty$ such that, as $t\to\infty$, In particular,

Figures (1)

  • Figure 1: Histogram of the standardized error $\sqrt{T}(\hat{\vartheta}_{T}-\vartheta)$ for $\vartheta=2.0, \mu=2.0, H=0.75$ with $T=100$. The red curve represents the theoretical asymptotic Gaussian density.

Theorems & Definitions (7)

  • Lemma 2.1
  • proof
  • Proposition 5.1
  • proof
  • Remark 6.1
  • Lemma 6.2
  • proof