Abstract
In this paper the filtering of partially observed diffusions, with discrete-time observations, 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 $\mathcal{O}(\varepsilon^2)$ mean-squared error between the multilevel estimator and average with respect to the filtering distribution is shown to scale optimally, for example, as $\mathcal{O}(\varepsilon^{-2})$ 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.