On a class of filtering problems on manifolds

The goal of the paper is to describe stochastic differential systems whose trajectories belong to a smooth manifold as an application to the optimal filtering problem. An additional condition is that not only system trajectories belong to the given manifold, but also the estimation results for these trajectories (solution of the optimal filtering problem with the minimum mean-squared error) belong to this manifold. Diffusion and jump-diffusion systems are considered. These systems can be driven by the Wiener process and the Poisson process. The main result is the conditions on coefficients of the equation for the estimated random process. These conditions are obtained on the basis of the first integral concept for the stochastic differential equation and some of its properties.