Kalman Filter and its Modern Extensions for the Continuous-time Nonlinear Filtering Problem
For researchers in filtering and stochastic control, this paper provides a unified review and a new algorithm, but the contribution is incremental as it builds on existing methods.
This paper reviews three algorithmic approaches for continuous-time nonlinear filtering: the Kalman-Bucy filter, the ensemble Kalman-Bucy filter (EnKBF), and the feedback particle filter (FPF). It also proposes a novel approximation algorithm based on optimal transport and coupling of measures, demonstrating its performance on a numerical example.
This paper is concerned with the filtering problem in continuous-time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter which provides an exact solution for the linear Gaussian problem, (ii) the ensemble Kalman-Bucy filter (EnKBF) which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems, and (iii) the feedback particle filter (FPF) which represents an extension of the EnKBF and furthermore provides for an consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of non-uniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.