Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes

The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–Uhlenbeck process defined as dXt=θXtdt+dGt,t≥0 with an unknown parameter θ>0θ>0, where GG is a Gaussian process. We provide sufficient conditions, based on the properties of GG, ensuring the strong consistency and the asymptotic distribution of our estimator θ˜t of θθ based on the observation {Xs,s∈[0,t]} as t→∞t→∞. Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when GG is a fractional Brownian motion with Hurst parameter H∈(0,1)H∈(0,1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.