| In this dissertation we consider two types of linear Stochastic Differential and Difference Equations (SDEs): Linear Homogeneous SDEs (LHSDEs) and the ARX models. Although LHSDEs form a relatively small subclass of SDEs, they are extremely important for many applications. LHSDEs can be used to describe a time evolution of a variety of physical systems. LHSDEs are a major tool for financial markets' modelling, and play the key role in pricing of derivative securities. There is an extensive engineering and mathematical literature on adaptive control of dynamic systems described by ARX model, due to their great success in industrial applications.;In this dissertation we reveal the exceptionally nice property of LHSDE that the ergodic and non-ergodic cases do not differ from the parameter estimation point of view: in both cases the Local Asymptotic Normality of the corresponding family of distributions with the normalizing factor T --1/2, and the asymptotic efficiency of the Maximum Likelihood Estimator is proved as the observation time T → infinity. It is shown that the Fisher's information matrix for the problem can be expressed in terms of the stationary distribution of an auxiliary Markov process on the unit sphere or the projective space PRd .;For the ARX models we consider the objective to find an adaptive control strategy enabling to estimate the parameters of the model as accurately as possible while keeping the output of the system below a specified level of variability. It turns out that this objective can be achieved by using the Astrom-Wittenmark self tuning tracker with a specific choice of the reference signal. It is shown that the Maximum Likelihood Estimator is asymptotically efficient for all control strategies of interest. |
