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

TL;DR

This paper develops a method for estimating the drift parameter in a partially observed mixed fractional Ornstein-Uhlenbeck process, extending previous models to include mixed fractional noise and analyzing the estimator's asymptotic properties.

Contribution

It introduces a new estimation framework for mixed fractional Ornstein-Uhlenbeck processes, including the construction of a Kalman filter and analysis of asymptotic normality.

Findings

MLE is consistent and asymptotically normal

Fisher Information matches the standard Brownian case

Extension to mixed fractional noise models

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.