A Bayes formula for Gaussian noise processes and its applications

An elementary approach is used to derive a Bayes-type formula, extending th e Kallianpur-Striebel formula for the nonlinear filters associated with the Gaussian noise processes. In the particular cases of certain Gaussian proc esses, recent results of Kunita and of Le Breton on fractional Brownian mot ion are derived. We also use the classical approximation of the Brownian mo tion by the Ornstein-Uhlenbeck dispersion process to solve the "instrumenta bility" problem of Balakrishnan. We give precise conditions for the converg ence of the filter based on the Ornstein-Uhlenbeck dispersion process to th e filter based on the Brownian motion. It is also shown that the solution o f the Zakai equation can be approximated by that of a ( deterministic) part ial differential equation.