  We present a particle filter construction for a system that exhibits
time-scale separation. The separation of time-scales allows two simplifications
that we exploit: i) The use of the averaging principle for the dimensional
reduction of the system needed to solve for each particle and ii) the
factorization of the transition probability which allows the
Rao-Blackwellization of the filtering step. Both simplifications can be
implemented using the coarse projective integration framework. The resulting
particle filter is faster and has smaller variance than the particle filter
based on the original system. The method is tested on a multiscale stochastic
differential equation and on a multiscale pure jump diffusion motivated by
chemical reactions.
