Langevin and Kalman Importance Sampling for Nonlinear Continuous-Discrete State-Space Models

The likelihood function of a nonlinear continuous-discrete state-space model with state dependent diffusion function is computed by integrating out the latent variables with the help of Langevin sampling. The continuous-time paths are discretized on a time grid in order to obtain a finite-dimensional integration and densities w.r.t. Lebesgue measure. We use importance sampling, where the exact importance density is the conditional density of the latent states, given the measurements. This unknown density is either estimated from the sampler data or approximated by an estimated normal density. Then, new trajectories are drawn from this Gaussian measure. Alternatively, a Gaussian importance density is directly derived from an extended Kalman smoother with subsequent sampling of independent trajectories (extended Kalman sampling (EKS)). We compare the Monte Carlo results with numerical methods based on extended, unscented, and Gauss-Hermite Kalman filtering (EKF, UKF, GHF) and a grid-based solution of the Fokker-Planck equation between measurements. This comprises the repeated multiplication of transition matrices based on Euler transition kernels, finite differences, and discretized integral operators. The methods are illustrated for the geometrical Brownian motion and the Ginzburg-Landau model for phase transitions.