Stochastic Differential Equation Models and State-Space Representations: Parameter Estimation Methods and Applications to Spatio-Temporal Systems

In this thesis, we explore methods for state space modelling, parameter estimation, and sparse spatio-temporal data processing within the framework of stochastic differ ential equation (SDE) models. After introducing the general form of SDEs, including the hypoelliptic case, we review key inference methods, including the splitting scheme, the discretization with the 1.5 scheme, and state-space methods based on the extended Kalman filter (EKF) and the Strang splitting Kalman filter (SSKF). We further extend this framework to models with time-varying parameters and conduct simulation studies on the Jansen–Rit model and the Stommel-Cessi two-box model. The core findings of this research are presented in three papers. The first paper demonstrates that, under various model structures, SDE models yield more robust parameter estimates than ODE models across multiple scenarios of model misspecification. The second paper applies a double-well SDE model with time-varying parameters to study the Atlantic Meridional Overturning Circulation (AMOC), revealing that carbon dioxide forcing is critical in determining tipping behaviour and the timing of collapse. The third paper models sparse spatial-temporal oceanographic data using East Greenland as a context. Our analysis reveals a shallowing and warming of the thermocline in the Scoresby Sound area, as well as a significant warming trend in the Greenland Sea, upporting the view of an ongoing ‘Atlantification’ process. Overall, this thesis shows the strengths of SDE-based modelling methods in capturing variability, improving inference robustness, and providing reliable insights into complex dynamic systems, while emphasising the necessity of using flexible methods in handling sparse data.