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

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.