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（SDEs）在系统科学、工程和生态科学等众多科学领域中作为强大的建模工具。尽管在给定问题中通常知道SDEs的具体形式，但某些模型参数仍然未知。基于离散时间序列的状态观测有效推断这些未知参数是一个重要的实际课题。非线性SDEs中面临挑战，由于状态转移概率密度函数和平稳概率密度函数缺乏闭式表达式，通常无法实现参数的最大似然估计。针对这一限制，我们提出了一种新颖的两步参数推断机制。该方法包括全局搜索阶段和局部精化过程。全局搜索阶段致力于识别高值似然函数的域，而局部精化过程专门设计用于在该局部域内增强替代似然。此外，我们提出了两种基于模拟的转移密度近似，旨在高效或精确近似似然函数。数值示例展示了我们提出的方法在实现后验参数估计中的有效性。