Modelling and parameter estimation for discretely observed fractional iterated Ornstein–Uhlenbeck processes

In this work we present how to model an observed time series by a FOU(p) process. We will show that the FOU(p) processes can be used to model a wide range of time series varying from short range dependence to long range dependence, with performance similar to the ARMA or ARFIMA models and in several cases outperforming them. Also, we extend the theoretical results for any FOU(p) processes for the case in which the Hurst parameter is less than 1/2 and we show theoretically and by simulations that under some conditions on T and the sample size n it is possible to obtain consistent estimators of the parameters when the process is observed in a discretized and equispaced interval [0,T]. Lastly, we give a way to obtain explicit formulas for the auto-covariance function for any FOU(p) and we present an application for FOU(2) and FOU(3).