Application of a resampling scheme to solve the divergence in the Pathwise filter

A Monte Carlo-based approach to filtering for nonlinear systems based on the Pathwise theory was proposed by M. H. A. Davis in 1981. The discrete-time Markov chain used to compute the solution of a Fokker-Planck equation whose coeficients were determined by the observed process was replaced by the simulation of a stochastic differential equation to make the filter implementation with less computational cost. This paper shows the Pathwise filter for an one-dimensional Ornstein-Uhlenbeck state process with saturation in the observation presents divergence in this estimates when the signal-to-noise ratio on the state equation is low. Rewriting the solution in terms of observation-based weights, it is presented the low performance of the filter can be explained by the increase of the weight variances. The problem is solved by a resampling scheme used to maintain the variance controlled.