Fast filtering with new sparse transition Markov chains

We put forward a novel Markov chain approximation method with regard to the filtering problem. The novelty consists in making use of the sparse grid theory to deal with the curse of dimensionality. Our method imitates the marginal distribution of the latent continuous process with a discrete probability distribution on a sparse grid. The grid points may be seen as the states of a Markov chain. We construct such a Markov chain to imitate the whole process. The transition probabilities are then chosen to preserve the joint moments of the underlying continuous process. We provide a simulation study of a multivariate stochastic volatility filtering problem where we compare the proposed methodology with a similar technique and with the particle filtering.