Maximum likelihood estimation in the ergodic Volterra Ornstein-Uhlenbeck process

We investigate maximum likelihood estimation for the drift parameters of stochastic Volterra processes in the ergodic regime. In our first result, we establish the equivalence of laws under general changes of drift and provide the corresponding Radon-Nikodym derivative. This allows us to develop a rigorous maximum likelihood estimation framework. As an application, we study the Volterra Ornstein–Uhlenbeck process in the ergodic regime, considering both continuous-time and high-frequency discrete-time observations. In both regimes, we prove the consistency and asymptotic normality of the maximum likelihood estimators. A key intermediate result, which may be of independent interest, is a uniform Birkhoff-type theorem under an asymptotic independence condition. This theorem yields a locally uniform Law of Large Numbers over the parameter space.