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 which deals 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 which we construct 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 on a multivariate stochastic volatility filtering problem to compare the proposed methodology with a similar technique and the particle filtering