Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X with drift coefficient b ( X t , α ) and diffusion coefficient ε a ( X t , 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 ε − 1 n − ρ 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.