Prof Valentine Genon-Catalot, Paris 5
Explicit filtering of discretized diffusions

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.  Concrete problems raised by these observations are the prediction, filtering or smoothing of (x_n).  This requires the computation of the conditional distributions of x_l given y_n, . . . , y_1, y_0 for all l, n.  We introduce sufficient conditions allowing to obtain explicit formulae for these conditional distributions and extend the notion of finite dimensional filters using mixtures of distributions.  The method is applied to the case where the signal x_n = Xn_ 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.