A note on inference for the mixed fractional Ornstein-Uhlenbeck process with drift

Abstract

This paper is devoted to the controlled drift estimation of the mixed fractional Ornstein-Uhlenbeck process. We will consider two models: one is the optimal input where we will find the controlled function which maximize the Fisher information for the unknown parameter and the other one with a constant as the controlled function. Large sample asymptotical properties of the Maximum Likelihood Estimator (MLE) is deduced using the Laplace transform computations or the Cameron-Martin formula with extra part from [ 12 ]. As a a supplement of [ 12 ] we will also prove that the MLE is strongly consistent.