Langevin and Kalman importance sampling for nonlinear continuous-discrete state space models
The likelihood function of a continuous-discrete state space model with state dependent diffusion function is computed by integrating out the latent variables with the help of a Langevin sampler. 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. We use 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.
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