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：本文旨在对离散观测扩散过程的参数推断领域作出系统性贡献。基于似然的传统估计方法需引入时间离散化方案，以近似随机微分方程模型在有限时间区间内难以处理的转移动态。该方案的步长既可人为预设，也可由数据特征决定。最新研究表明，数值方案的选择对次椭圆随机微分方程参数估计量的行为具有关键影响。具体而言，本研究首先构建两类弱二阶抽样方案（分别适用于次椭圆与椭圆型随机微分方程），并导出这些方案密度的小时间展开式，作为真实不可处理转移密度的代理。随后，我们针对基于这些代理似然获得的参数估计量建立一套解析结果，从而构建理论框架以彰显所提方法论在随机微分方程校准中的优势。最后，通过经典推断与贝叶斯推断两种范式，呈现针对椭圆与次椭圆随机微分方程的数值实验成果。