Simulation-based transition density approximation for the inference of SDE models

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.