This work aims at making a comprehensive contribution in the general area of parametric inference for discretely observed diffusion processes. Established approaches for likelihood-based estimation invoke a time-discretisation scheme for the approximation of the intractable transition dynamics of the Stochastic Differential Equation (SDE) model over finite time periods. The scheme is applied for a step-size that is either user-selected or determined by the data. Recent research has highlighted the critical ef-fect of the choice of numerical scheme on the behaviour of derived parameter estimates in the setting of hypo-elliptic SDEs. In brief, in our work, first, we develop two weak second order sampling schemes (to cover both hypo-elliptic and elliptic SDEs) and produce a small time expansion for the density of the schemes to form a proxy for the true intractable SDE transition density. Then, we establish a collection of analytic results for likelihood-based parameter estimates obtained via the formed proxies, thus providing a theoretical framework that showcases advantages from the use of the developed methodology for SDE calibration. We present numerical results from carrying out classical or Bayesian inference, for both elliptic and hypo-elliptic SDEs.

翻译：本工作旨在对离散观测扩散过程的参数推断领域做出全面贡献。已有基于似然的估计方法通过对随机微分方程在有限时间段内的难处理转移动力学进行时间离散化近似。该离散化方案采用用户自选或由数据决定的步长。近期研究凸显了数值方案选择对次椭圆随机微分方程参数估计行为的关键影响。具体而言，本文首先开发了两种弱二阶采样方案（覆盖次椭圆和椭圆随机微分方程），并推导出方案密度的短时展开式，作为真实难处理随机微分方程转移密度的近似代理。随后，我们建立了通过所构建代理获得的基于似然的参数估计分析结果体系，从而为展示所开发方法在随机微分方程校准中的优势提供了理论框架。我们对椭圆和次椭圆随机微分方程分别展示了经典推断和贝叶斯推断的数值结果。