Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X with drift coefficient b(Xt, α) and diffusion coefficient εa(Xt, β) where α and β are two unknown parameters, while ε 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 (α, β). The estimators are shown to be consistent and asymptotically normal when n → ∞ and ε → 0 in such a way that εn remains bounded for some ρ > 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.