On Filtering with Observation in a Manifold: Reduction to a Classical Filtering Problem

The current paper deals with filtering problems where the observation process, conditioned on the unknown signal, is an elliptic diffusion in a differentiable manifold. Precisely, the observation model is given by a stochastic differential equation in the underlying manifold. The main new idea is to use a Le Jan-Watanabe connection, instead of a usual Levi-Civita connection, in performing the operation of antidevelopment (both connections can be constructed from a stochastic differential equation as in the observation model). The following results are obtained. First, it is shown that antidevelopment reduces the original filtering problem to a classical one, i.e. an additive white noise model. Second, a new form of the Zakai and filtering equations is derived which has a structure very similar to that of classical equations. Currently, there are few well understood general numerical solution methods for filtering problems with observation in a manifold. Reduction to classical filtering problems seems desirable as it allows proven existing numerical methods for classical problems to be applied immediately. In view of practical implementation of this approach, the paper also gives an explicit pathwise construction of the antidevelopment.