Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X with drift coefficient b(Xt,[alpha]) and diffusion coefficient [epsilon]a(Xt,[beta]) where [alpha] and [beta] are two unknown parameters, while [epsilon] is known. For a high frequency sample of observations of the diffusion at the time points k/n, k=1,...,n, we propose a class of contrast functions and thus obtain estimators of ([alpha],[beta]). The estimators are shown to be consistent and asymptotically normal when n-->[infinity] and [epsilon]-->0 in such a way that [epsilon]-1n-[rho] remains bounded for some [rho]>0. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.