A contrast estimator for completely or partially observed hypoelliptic diffusion

Parametric estimation of two-dimensional hypoelliptic diffusions is considered when complete observations–both coordinates discretely observed–or partial observations–only one coordinate observed–are available. Since the volatility matrix is degenerate, Euler contrast estimators cannot be used directly. For complete observations, we introduce an Euler contrast based on the second coordinate only. For partial observations, we define a contrast based on an integrated diffusion resulting from a transformation of the original one. A theoretical study proves that the estimators are consistent and asymptotically Gaussian. A numerical application to Langevin systems illustrates the nice properties of both complete and partial observations’ estimators.