A Bayes formula for Gaussian noise processes and its applications

An elementary approach is used to derive a Bayes-type formula, extending the Kallianpur--Striebel formula for the nonlinear filters associated with the Gaussian noise processes. In the particular cases of certain Gaussian processes, recent results of Kunita and of Le Breton on fractional Brownian motion are derived. We also use the classical approximation of the Brownian motion by the Ornstein--Uhlenbeck dispersion process to solve the "instrumentability" problem of Balakrishnan. We give precise conditions for the convergence of the filter based on the Ornstein--Uhlenbeck dispersion process to the filter based on the Brownian motion. It is also shown that the solution of the Zakai equation can be approximated by that of a (deterministic) partial differential equation