Multilevel particle filters

In this paper the filtering of partially observed diffusions, with discrete-time observa- tions, is considered. It is assumed that only biased approximations of the diffusion can be obtained for choice of an accuracy parameter indexed by l. A multilevel estimator is proposed consisting of a telescopic sum of increment estimators associated to the successive levels. The work associated to O("ϵ2) mean-squared error between the multilevel estimator and average with respect to the filtering distribution is shown to scale optimally, for example, as O("ϵ2) for optimal rates of convergence of the underlying diffusion approximation. The method is illustrated with some toy examples as well as estimation of interest rate based on real S&P 500 stock price data.