We study parametric inference for ergodic diffusion processes with a degenerate diffusion matrix. Existing research focuses on a particular class of hypo-elliptic SDEs, with components split into `rough'/`smooth' and noise from rough components propagating directly onto smooth ones, but some critical model classes arising in applications have yet to be explored. We aim to cover this gap, thus analyse the highly degenerate class of SDEs, where components split into further sub-groups. Such models include e.g. the notable case of generalised Langevin equations. We propose a tailored time-discretisation scheme and provide asymptotic results supporting our scheme in the context of high-frequency, full observations. The proposed discretisation scheme is applicable in much more general data regimes and is shown to overcome biases via simulation studies also in the practical case when only a smooth component is observed. Joint consideration of our study for highly degenerate SDEs and existing research provides a general `recipe' for the development of time-discretisation schemes to be used within statistical methods for general classes of hypo-elliptic SDEs.

翻译：我们研究了具有退化扩散矩阵的遍历扩散过程的参数推断。现有研究主要关注一类特定的亚椭圆随机微分方程（SDE），其分量被分为“粗糙”/“光滑”两类，且粗糙分量的噪声直接传播到光滑分量上，但应用中出现的某些关键模型类别尚未被探索。本文旨在填补这一空白，因此我们分析了分量可进一步划分为子组的高度退化SDE类。此类模型包括广义朗之万方程等典型案例。我们提出了一种量身定制的时间离散化方案，并基于高频全观测数据提供了支持该方案的渐近理论结果。所提出的离散化方案适用于更广泛的数据场景，并在仅观测到光滑分量的实际情况下通过仿真研究证明了其能克服偏差。本文将我们对高度退化SDE的研究与现有研究相结合，为统计方法中针对一般类亚椭圆SDE开发时间离散化方案提供了通用“配方”。