Computable infinite-dimensional filters with applications to discretized diffusion processes

Let us consider a pair signal–observation ( ( x n , y n ) , n ≥ 0 ) where the unobserved signal ( x n ) is a Markov chain and the observed component is such that, given the whole sequence ( x n ) , the random variables ( y n ) are independent and the conditional distribution of y n only depends on the corresponding state variable x n . The main problems raised by these observations are the prediction and filtering of ( x n ) . We introduce sufficient conditions allowing us to obtain computable filters using mixtures of distributions. The filter system may be finite or infinite-dimensional. The method is applied to the case where the signal x n = X n Δ is a discrete sampling of a one-dimensional diffusion process: Concrete models are proved to fit in our conditions. Moreover, for these models, exact likelihood inference based on the observation ( y 0 , … , y n ) is feasible.