Stochastic Differential Equations (SDEs) serve as a powerful modeling tool in various scientific domains, including systems science, engineering, and ecological science. While the specific form of SDEs is typically known for a given problem, certain model parameters remain unknown. Efficiently inferring these unknown parameters based on observations of the state in discrete time series represents a vital practical subject. The challenge arises in nonlinear SDEs, where maximum likelihood estimation of parameters is generally unfeasible due to the absence of closed-form expressions for transition and stationary probability density functions of the states. In response to this limitation, we propose a novel two-step parameter inference mechanism. This approach involves a global-search phase followed by a local-refining procedure. The global-search phase is dedicated to identifying the domain of high-value likelihood functions, while the local-refining procedure is specifically designed to enhance the surrogate likelihood within this localized domain. Additionally, we present two simulation-based approximations for the transition density, aiming to efficiently or accurately approximate the likelihood function. Numerical examples illustrate the efficacy of our proposed methodology in achieving posterior parameter estimation.

翻译：随机微分方程（SDE）作为系统科学、工程学及生态学等多个科学领域的重要建模工具，其具体形式在特定问题中通常已知，但部分模型参数仍属未知。如何基于离散时间序列的状态观测高效推断这些未知参数，是一个具有重要实用价值的课题。在非线性SDE中，由于状态转移密度和平稳概率密度函数缺乏闭式表达式，通常无法直接进行参数的最大似然估计。针对这一局限，我们提出了一种新颖的两步参数推断机制，包含全局搜索阶段与局部精化步骤。全局搜索阶段致力于识别高似然函数值的定义域，而局部精化步骤则专门用于在该局部区域内提升代理似然的精度。此外，我们提出了两种基于仿真的转移密度近似方法，旨在高效或精确地逼近似然函数。数值实验验证了所提方法在实现参数后验估计中的有效性。