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 coefficients were determined by the observed process was here replaced by the simulation of an equivalent stochastic differential equation in order to make the filter implementation more clear and with less computational cost. This paper shows that the Pathwise filter for an one-dimensional Ornstein-Uhlenbeck state process with saturation in the observation has an interesting characteristic of divergence in this estimates when the signal-to-noise ratio on the state equation is low. Rewriting the filtering solution in terms of observation-based weights, it is presented that the low performance of the filter can be preliminary explained by the sudden increase of the weight variances. To solve this problem, a resampling scheme using the effective number of particles was used to smooth or, at least, maintain the weight variance controlled.