Parameter Estimation for a partially observed Ornstein-Uhlenbeck process with long-memory noise

Abstract

\noindent \textbf{Abstract}: We consider the parameter estimation problem for the Ornstein-Uhlenbeck process $X$ driven by a fractional Ornstein-Uhlenbeck process $V$, i.e. the pair of processes defined by the non-Markovian continuous-time long-memory dynamics $dX_{t}=-θX_{t}dt+dV_{t};\ t\geq 0$, with $dV_{t}=-ρV_{t}dt+dB_{t}^{H};\ t\geq 0$, where $θ>0$ and $ρ>0$ are unknown parameters, and $B^{H}$ is a fractional Brownian motion of Hurst index $H\in (\frac{1}{2},1)$. We study the strong consistency as well as the asymptotic normality of the joint least squares estimator $(\hatθ_{T},\widehat{ρ}% _{T}) $ of the pair $( θ,ρ) $, based either on continuous or discrete observations of $\{X_{s};\ s\in \lbrack 0,T]\}$ as the horizon $T$ increases to +$\infty $. Both cases qualify formally as partial-hbobservation questions since $V$ is unobserved. In the latter case, several discretization options are considered. Our proofs of asymptotic normality based on discrete data, rely on increasingly strict restrictions on the sampling frequency as one reduces the extent of sources of observation. The strategy for proving the asymptotic properties is to study the case of continuous-time observations using the Malliavin calculus, and then to exploit the fact that each discrete-data estimator can be considered as a perturbation of the continuous one in a mathematically precise way, despite the fact that the implementation of the discrete-time estimators is distant from the continuous estimator. In this sense, we contend that the continuous-time estimator cannot be implemented in practice in any naïve way, and serves only as a mathematical tool in the study of the discrete-time estimators' asymptotics.