  We consider parametric estimation and tests for multi-dimensional diffusion
processes with a small dispersion parameter $\varepsilon$ from discrete
observations. For parametric estimation of diffusion processes, the main target
is to estimate the drift parameter and the diffusion parameter. In this paper,
we propose two types of adaptive estimators for both parameters and show their
asymptotic properties under $\varepsilon\to0$, $n\to\infty$ and the balance
condition that $(\varepsilon n^\rho)^{-1} =O(1)$ for some $\rho>0$. Using these
adaptive estimators, we also introduce consistent adaptive testing methods and
prove that test statistics for adaptive tests have asymptotic distributions
under null hypothesis. In simulation studies, we examine and compare asymptotic
behaviors of the two kinds of adaptive estimators and test statistics.
Moreover, we treat the SIR model which describes a simple epidemic spread for a
biological application.
