Monte-Carlo methods in nonlinear filtering and importance sampling

For the calculation of conditional expectations in nonlinear filtering of Markov processes, one may think to use Monte-Carlo techniques, as an alternative to the numerical solution of Zakai equation (a stochastic PDE). We show that a direct implementation of this idea is unefficient, and we propose a modified algorithm, that uses importance sampling, where our choice of the new probability is based on large deviations arguments.