Analyse statistique de quelques modèles de processus de type fractionnaire

This thesis focuses on the statistical analysis of some models of stochastic processes generated by fractional noise in discrete or continuous time.In Chapter 1, we study the problem of parameter estimation by maximum likelihood (MLE) for an autoregressive process of order p (AR (p)) generated by a stationary Gaussian noise, which can have long memory as the fractional Gaussiannoise. We exhibit an explicit formula for the MLE and we analyze its asymptotic properties. Actually in our model the covariance function of the noise is assumed to be known but the asymptotic behavior of the estimator ( rate of convergence, Fisher information) does not depend on it.Chapter 2 is devoted to the determination of the asymptotical optimal input for the estimation of the drift parameter in a partially observed but controlled fractional Ornstein-Uhlenbeck process. We expose a separation principle that allows us toreach this goal. Large sample asymptotical properties of the MLE are deduced using the Ibragimov-Khasminskii program and Laplace transform computations for quadratic functionals of the process.In Chapter 3, we present a new approach to study the properties of mixed fractional Brownian motion (fBm) and related models, based on the filtering theory of Gaussian processes. The results shed light on the semimartingale structure andproperties lead to a number of useful absolute continuity relations. We establish equivalence of the measures, induced by the mixed fBm with stochastic drifts, and derive the corresponding expression for the Radon-Nikodym derivative. For theHurst index H > 3=4 we obtain a representation of the mixed fBm as a diffusion type process in its own filtration and derive a formula for the Radon-Nikodym derivative with respect to the Wiener measure. For H 3=4, nous obtenons une représentation du mouvement brownien fractionnaire mélangé comme processus de type diffusion dans sa filtration naturelle et en déduisons une formule de la dérivée de Radon-Nikodym par rapport à la mesurede Wiener. Pour H < 1=4, nous montrons l’équivalence de la mesure avec celle la composante fractionnaire et obtenons une formule pour la densité correspondante. Un domaine d’application potentielle est l’analyse statistique des modèles gouvernés par des bruits fractionnaires mélangés. A titre d’exemple, nous considérons le modèle de régression linéaire de base et montrons comment définir l’EMV et étudié son comportement asymptotique