Classification by continuous-time observations in multiplicative noise I: formulae for Bayesian estimate

The two-part paper is devoted to the estimation of a finite-state random vector given the continuous-time noised observations. The key feature is that the observation noise intensity is a function of the estimated vector that makes useless the known results in the optimal filtering. The estimate is obtained both in the explicit integral form and as a solution to a stochastic differential system with some jump processes in the right-hand side.