Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion XX with drift coefficient b(Xt,α)b(Xt,α) and diffusion coefficient εa(Xt,β)εa(Xt,β) where αα and ββ are two unknown parameters, while εε is known. For a high frequency sample of observations of the diffusion at the time points k/nk/n, k=1,…,nk=1,…,n, we propose a class of contrast functions and thus obtain estimators of (α,β)(α,β). The estimators are shown to be consistent and asymptotically normal when n→∞n→∞ and ε→0ε→0 in such a way that ε−1n−ρε−1n−ρ remains bounded for some ρ>0ρ>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.