Estimation of an Ergodic Diffusion from Discrete Observations

We consider a one-dimensional diffusion process X , with ergodic property, with drift b ( x , θ) and diffusion coefficient a ( x , σ) depending on unknown parameters θ and σ. We are interested in the joint estimation of (θ, σ). For that purpose, we dispose of a discretized trajectory, observed at n equidistant times tⁿ_i = ih_n , 1 ≤ i ≤ n . We assume that h_n ← 0 and nh_n ←∞. Under the condition nh_n^p ← 0 for an arbitrary integer p , we exhibit a contrast dependent on p which provides us with an asymptotically normal and efficient estimator of (θ, σ).