Introdution

Abstract

In this work, we are interested in the problems of estimation theory concerned with observations of the diffusion-type process

$$d{{X}_{t}} = {{S}_{t}}\left( X \right)dt + \varepsilon dWt,{{X}_{0}} = {{x}_{0}},0 \leqslant t \leqslant T,$$

(0.1)

Where W_t is a standard Wiener process and S_t(·)is some nonanticipative smooth function. By the observations ${\text{X}} = \left\{ {{{X}_{t}},0 \leqslant t \leqslant T} \right\}$ of this process, we will solve some of the problems of identification, both parametric and nonparametric. If the trend S(·) is known up to the value of some finite-dimensional parameter ${{S}_{t}}\left( X \right) = {{S}_{t}}\left( {0,X} \right),$ $\theta \in \Theta \subset {{R}^{d}}$, then we have a parametric case. The nonparametric problems arise if we know only the degree of smoothness of the function ${{S}_{t}}\left( X \right),0 \leqslant t \leqslant T$ with respect to time t. It is supposed that the diffusion coefficient E is always known.in the parametric case, we describe the asymptotical properties of maximum likelihood (MLE), Bayes (BE) and minimum distance (MDE) estimators $\varepsilon \to 0$ and in the nonparametric situation, we investigate some kernel-type estimators of unknown functions (say, S _t (•), $0 \leqslant t \leqslant T$).