On estimating the drift of a diffusion process from observations in a domain

Let $x_t$ be a diffusion process in a domain $G$ of a Euclidian space with small diffusion proportional to $\varepsilon$ and drift depending on an unknown parameter $\theta$. In this paper, a lower bound for $\mathbf M_{\theta,x}\|\theta-\theta_1\|^2$ is obtained (see (0.6)), where $\theta_1$ is an arbitrary estimate, of $\theta$, and locally asymptotically minimax estimates are found.