  We consider the problem of statistical inference for the effective dynamics
of multiscale diffusion processes with (at least) two widely separated
characteristic time scales. More precisely, we seek to determine parameters in
the effective equation describing the dynamics on the longer diffusive time
scale, i.e. in a homogenization framework. We examine the case where both the
drift and the diffusion coefficients in the effective dynamics are
space-dependent and depend on multiple unknown parameters. It is known that
classical estimators, such as Maximum Likelihood and Quadratic Variation of the
Path Estimators, fail to obtain reasonable estimates for parameters in the
effective dynamics when based on observations of the underlying multiscale
diffusion. We propose a novel algorithm for estimating both the drift and
diffusion coefficients in the effective dynamics based on a semi-parametric
framework. We demonstrate by means of extensive numerical simulations of a
number of selected examples that the algorithm performs well when applied to
data from a multiscale diffusion. These examples also illustrate that the
algorithm can be used effectively to obtain accurate and unbiased estimates.
