Inference in stochastic differential equations using the Laplace approximation: Demonstration and examples

We consider the problem of estimating states and parameters in a model based on a system of coupled stochastic differential equations, based on noisy discrete-time data. Special attention is given to nonlinear dynamics and state-dependent diffusivity, where transition densities are not available in closed form. Our technique adds states between times of observations, approximates transition densities using, e.g., the Euler-Maruyama method and eliminates unobserved states using the Laplace approximation. Using case studies, we demonstrate that transition probabilities are well approximated, and that inference is computationally feasible. We discuss limitations and potential extensions of the method.