We address the problem of parameter estimation for degenerate diffusion processes defined via the solution of Stochastic Differential Equations (SDEs) with diffusion matrix that is not full-rank. For this class of hypo-elliptic diffusions recent works have proposed contrast estimators that are asymptotically normal, provided that the step-size in-between observations $\Delta=\Delta_n$ and their total number $n$ satisfy $n \to \infty$, $n \Delta_n \to \infty$, $\Delta_n \to 0$, and additionally $\Delta_n = o (n^{-1/2})$. This latter restriction places a requirement for a so-called `rapidly increasing experimental design'. In this paper, we overcome this limitation and develop a general contrast estimator satisfying asymptotic normality under the weaker design condition $\Delta_n = o(n^{-1/p})$ for general $p \ge 2$. Such a result has been obtained for elliptic SDEs in the literature, but its derivation in a hypo-elliptic setting is highly non-trivial. We provide numerical results to illustrate the advantages of the developed theory.

翻译：本文研究退化扩散过程的参数估计问题，这类过程由扩散矩阵非满秩的随机微分方程(SDEs)的解定义。对于此类次椭圆扩散，最新研究提出了对比估计量，该估计量在观测步长$\Delta=\Delta_n$与观测总数$n$满足$n \to \infty$、$n \Delta_n \to \infty$、$\Delta_n \to 0$且额外满足$\Delta_n = o (n^{-1/2})$时具有渐近正态性。后一约束条件要求采用所谓的"快速递增实验设计"。本文突破了这一限制，在更弱的设计条件$\Delta_n = o(n^{-1/p})$（其中$p \ge 2$为一般参数）下，建立了满足渐近正态性的通用对比估计量。该结果已在椭圆SDEs文献中获得，但在次椭圆框架下的推导具有高度非平凡性。我们通过数值结果展示了所发展理论的优势。