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

TL;DR

This work develops a Laplace-approximation-based framework for estimating states and parameters in nonlinear stochastic differential equations with state-dependent diffusion from noisy discrete-time data. By inserting intermediate time points and either modeling Brownian increments or working in state-space, the method yields tractable approximations to transition densities and enables efficient joint inference. It provides both Itô and Stratonovich formulations, analyzes biases of naive approaches, and demonstrates practical performance on CIR transition densities and a stochastic predator–prey model. The approach is flexible, implementable with existing tools, and applicable to non-additive noise, with work planned on continuous-time limits and accuracy analyses.

Abstract

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.

Figures (6)

• Figure 1: Simulation and re-estimation in the Ornstein-Uhlenbeck process; the present paper allows generalization of this example to non-Gaussian situations. Solid: The simulated sample path. Dots: Simulated observed values with measurement noise. Dashed: The estimated path. Grey zone: Marginal 68 % confidence intervals, constructed as estimate $\pm$ one standard deviation.
• Figure 2: The Geometric Brownian Bridge: Modes of the finite-dimensional distributions for various numbers of interpolating points.
• Figure 3: Probabilistic graphical network of random variables in a diffusion process $\{X_t\}$ driven by increments of the underlying Brownian motion $\{B_t\}$.
• Figure 4: Transition probabilities for the CIR process in natural scale (top panel) and log scale (bottom panel): Comparison between the analytical densities (circles), and the densities computed using the Laplace approximation using the four different methods, which give practically indistinguishable results. See the text for parameters and full description.
• Figure 5: Errors in the transition probabilities for the CIR process: Absolute errors $p-p_A$ (top panel) and relative errors $p/p_A - 1$ (bottom panel), for the four different methods. The curves for the Itô-based methods "dB", "XdB", and "X" are practically indistinguishable. See the text for parameters and full description.