Statistical analysis of some fractional process models.

Summary In Chapter 1, we study the maximum likelihood estimation (MLE) problem of the parameters of a p-order autoregressive process (AR(p)) driven by a stationary Gaussian noise, which can be long-memory like the fractional Gaussian noise. We give an explicit formula for the MLE and analyze its asymptotic properties. In fact, in our model the covariance function of the noise is assumed to be known, but the asymptotic behavior of the estimator (speed of convergence, Fisher information) does not depend on it.Chapter 2 is devoted to the determination of the optimal input (from an asymptotic point of view) for the estimation of the drift parameter in a partially observed but controlled fractional Ornstein-Uhlenbeck process. We present a separation principle that allows us to achieve this goal. The asymptotic properties of the MLE are demonstrated using the Ibragimov-Khasminskii program and the computation of Laplace transforms of a quadratic functional of the process.In Chapter 3, we present a new approach to study the properties of the mixed fractional Brownian motion and related models, based on the theory of filtering Gaussian processes. The results highlight the semimartingale structure and lead to a number of useful continuity absolute properties. We establish the equivalence of measures induced by mixed fractional Brownian motion with a stochastic drift, and derive the corresponding expression for the Radon-Nikodym derivative. For a Hurst index H > 3=4, we obtain a representation of the mixed fractional Brownian motion as a diffusion-like process in its natural filtration and deduce a formula for the Radon-Nikodym derivative with respect to the Wiener measure. For H < 1=4, we show the equivalence of the measure with that of the fractional component and obtain a formula for the corresponding density. A potential field of application is the statistical analysis of models governed by mixed fractional noises. As an example, we consider the basic linear regression model and show how to define the MLE and study its asymptotic behavior.