We study parametric inference for hypo-elliptic Stochastic Differential Equations (SDEs). 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：我们研究欠椭圆随机微分方程（Stochastic Differential Equations, SDEs）的参数推断问题。现有研究聚焦于特定类别的欠椭圆SDEs，其分量划分为“粗糙”/“光滑”类型，且噪声从粗糙分量直接传播至光滑分量，但应用中出现的若干关键模型类别尚未被探索。为填补这一空白，我们分析了高度退化的SDEs类别，其中分量可进一步细分为子组。此类模型包括广义Langevin方程等典型案例。我们提出了一种定制化的时间离散化方案，并提供了在高频、完全观测场景下支撑该方案的渐近结果。所提出的离散化方案适用于更广泛的数据模式，并通过仿真研究证明其能在仅观测光滑分量的实际情形下克服偏差。将本研究对高度退化SDEs的分析与现有研究相结合，可为针对一般类别欠椭圆SDEs的统计方法中时间离散化方案的开发提供通用“配方”。