Consider a diffusion X=(Xt,t≥0) in Rd which is a solution of (i) dXt=f(θ,Xt)dt+dWt, X0=x0, and the deterministic analogue (ii) dU(θ,t)=f(θ,U(θ,t))dt, U(θ,t0)=x0. We shall prove that when f satisfies the usual Lipschitz and growth conditions, the solutins of (i) and (ii) are closed in some sense. Then we use this result to prove consistency of a least squares type estimator of the parameter θ. A strong consistency result will be proved for a special case of model (i). Finally, simulated observations will be used to compare our estimator with the one proposed by Dorogovcev and Prakasa Rao.
